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Quantum Graph API

kmeanssa_ng.quantum_graph

Quantum graph implementation and utilities.

MinimizeEnergyNodeUpdate

Bases: LloydUpdateStrategy

Update strategy that finds the node that minimizes k-means energy.

This strategy first maps each point in the cluster to its nearest node, then finds the graph node that minimizes the sum of squared distances (k-means energy) to this set of cluster nodes.

Source code in kmeanssa_ng/quantum_graph/lloyd_update.py 
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class MinimizeEnergyNodeUpdate(LloydUpdateStrategy):
    """Update strategy that finds the node that minimizes k-means energy.

    This strategy first maps each point in the cluster to its nearest node,
    then finds the graph node that minimizes the sum of squared distances
    (k-means energy) to this set of cluster nodes.
    """

    def __init__(self, random_state: int | np.random.Generator | None = None):
        # Normalize once (see MostFrequentNodeUpdate): one advancing stream
        # across update() calls rather than a fresh default_rng each time.
        self._rng = (
            random_state
            if isinstance(random_state, np.random.Generator)
            else np.random.default_rng(random_state)
        )

    def update(self, points: list[QGPoint], space: "QuantumGraph") -> "QGCenter":
        """Compute the new center by finding the node that minimizes energy.

        Args:
            points: A list of points belonging to a single cluster.
            space: The quantum graph, which must have distances precomputed.

        Returns:
            The new center for the cluster.
        """
        if not points:
            return None

        if space._pairwise_nodes_distance_array is None:
            raise ValueError("MinimizeEnergyNodeUpdate requires precomputed distances.")

        # 1. Replace each point with its closest node
        cluster_nodes = [p.closest_node() for p in points]

        # 2. Find the node that minimizes the energy
        dist_matrix = space._pairwise_nodes_distance_array
        node_to_idx = space._node_to_index
        all_graph_nodes = list(space.nodes())

        cluster_node_indices = [node_to_idx[n] for n in cluster_nodes]

        min_energy = float("inf")
        best_node = -1

        # Iterate through all nodes of the graph as potential centers
        for i, candidate_node in enumerate(all_graph_nodes):
            candidate_node_idx = i  # Index corresponds to order in all_graph_nodes

            # Calculate energy for this candidate node
            energy = 0.0
            for node_idx in cluster_node_indices:
                energy += dist_matrix[candidate_node_idx, node_idx] ** 2

            if energy < min_energy:
                min_energy = energy
                best_node = candidate_node

        if best_node == -1:
            # This should not happen if there are points
            raise ValueError("Could not determine best node to minimize energy.")

        # 3. Return the new center
        return space.node_as_center(best_node, rng=self._rng)

update(points, space)

Compute the new center by finding the node that minimizes energy.

Parameters:

Name Type Description Default
points list[QGPoint]

A list of points belonging to a single cluster.

 required
space 'QuantumGraph'

The quantum graph, which must have distances precomputed.

 required

Returns:

Type Description
'QGCenter'

The new center for the cluster.
Source code in kmeanssa_ng/quantum_graph/lloyd_update.py 
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def update(self, points: list[QGPoint], space: "QuantumGraph") -> "QGCenter":
    """Compute the new center by finding the node that minimizes energy.

    Args:
        points: A list of points belonging to a single cluster.
        space: The quantum graph, which must have distances precomputed.

    Returns:
        The new center for the cluster.
    """
    if not points:
        return None

    if space._pairwise_nodes_distance_array is None:
        raise ValueError("MinimizeEnergyNodeUpdate requires precomputed distances.")

    # 1. Replace each point with its closest node
    cluster_nodes = [p.closest_node() for p in points]

    # 2. Find the node that minimizes the energy
    dist_matrix = space._pairwise_nodes_distance_array
    node_to_idx = space._node_to_index
    all_graph_nodes = list(space.nodes())

    cluster_node_indices = [node_to_idx[n] for n in cluster_nodes]

    min_energy = float("inf")
    best_node = -1

    # Iterate through all nodes of the graph as potential centers
    for i, candidate_node in enumerate(all_graph_nodes):
        candidate_node_idx = i  # Index corresponds to order in all_graph_nodes

        # Calculate energy for this candidate node
        energy = 0.0
        for node_idx in cluster_node_indices:
            energy += dist_matrix[candidate_node_idx, node_idx] ** 2

        if energy < min_energy:
            min_energy = energy
            best_node = candidate_node

    if best_node == -1:
        # This should not happen if there are points
        raise ValueError("Could not determine best node to minimize energy.")

    # 3. Return the new center
    return space.node_as_center(best_node, rng=self._rng)

MostFrequentNode

Bases: RobustificationStrategy[list[Any]]

Strategy to find the most frequent node for each center.

Returns QGCenter objects located at the most frequently visited nodes during the robustification phase.

Source code in kmeanssa_ng/quantum_graph/robustification.py 
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class MostFrequentNode(RobustificationStrategy[list[Any]]):
    """Strategy to find the most frequent node for each center.

    Returns QGCenter objects located at the most frequently visited nodes
    during the robustification phase.
    """

    def initialize(self, sa: SimulatedAnnealing) -> None:
        """Initialize an empty list to store node collections.

        Raises:
            TypeError: If the space is not a QuantumGraph.
        """
        from .space import QuantumGraph

        if not isinstance(sa.space, QuantumGraph):
            raise TypeError(
                "MostFrequentNode strategy can only be used with QuantumGraph spaces."
            )

        self._central_nodes_collections: list[list] = []
        self.sa = sa

    def collect(self, sa: SimulatedAnnealing) -> None:
        """Collect the closest node for each center at the current step."""
        current_nodes = [center.closest_node() for center in sa.centers]
        self._central_nodes_collections.append(current_nodes)

    def get_result(self) -> list[Any]:
        """Return QGCenter objects at the most frequent nodes.

        Always returns a list of QGCenter objects, even for k=1.
        """
        if not self._central_nodes_collections:
            return []

        num_centers = len(self._central_nodes_collections[0])
        transposed_nodes = [
            [nodes[i] for nodes in self._central_nodes_collections]
            for i in range(num_centers)
        ]

        robust_nodes = [
            Counter(center_nodes).most_common(1)[0][0]
            for center_nodes in transposed_nodes
        ]

        # Convert node IDs to QGCenter objects
        from .space import QuantumGraph

        if isinstance(self.sa.space, QuantumGraph):
            robust_centers = [
                self.sa.space.node_as_center(node, rng=self.sa._rng)
                for node in robust_nodes
            ]
        else:
            # Fallback for non-QuantumGraph spaces (shouldn't happen with this strategy)
            robust_centers = robust_nodes

        # Always return a list for consistency
        return robust_centers

collect(sa)

Collect the closest node for each center at the current step.

Source code in kmeanssa_ng/quantum_graph/robustification.py 
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def collect(self, sa: SimulatedAnnealing) -> None:
    """Collect the closest node for each center at the current step."""
    current_nodes = [center.closest_node() for center in sa.centers]
    self._central_nodes_collections.append(current_nodes)

get_result()

Return QGCenter objects at the most frequent nodes.

Always returns a list of QGCenter objects, even for k=1.

Source code in kmeanssa_ng/quantum_graph/robustification.py 
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def get_result(self) -> list[Any]:
    """Return QGCenter objects at the most frequent nodes.

    Always returns a list of QGCenter objects, even for k=1.
    """
    if not self._central_nodes_collections:
        return []

    num_centers = len(self._central_nodes_collections[0])
    transposed_nodes = [
        [nodes[i] for nodes in self._central_nodes_collections]
        for i in range(num_centers)
    ]

    robust_nodes = [
        Counter(center_nodes).most_common(1)[0][0]
        for center_nodes in transposed_nodes
    ]

    # Convert node IDs to QGCenter objects
    from .space import QuantumGraph

    if isinstance(self.sa.space, QuantumGraph):
        robust_centers = [
            self.sa.space.node_as_center(node, rng=self.sa._rng)
            for node in robust_nodes
        ]
    else:
        # Fallback for non-QuantumGraph spaces (shouldn't happen with this strategy)
        robust_centers = robust_nodes

    # Always return a list for consistency
    return robust_centers

initialize(sa)

Initialize an empty list to store node collections.

Raises:

Type Description
TypeError

If the space is not a QuantumGraph.
Source code in kmeanssa_ng/quantum_graph/robustification.py 
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def initialize(self, sa: SimulatedAnnealing) -> None:
    """Initialize an empty list to store node collections.

    Raises:
        TypeError: If the space is not a QuantumGraph.
    """
    from .space import QuantumGraph

    if not isinstance(sa.space, QuantumGraph):
        raise TypeError(
            "MostFrequentNode strategy can only be used with QuantumGraph spaces."
        )

    self._central_nodes_collections: list[list] = []
    self.sa = sa

MostFrequentNodeUpdate

Bases: LloydUpdateStrategy

Update strategy that computes the new center as the most frequent closest node to the points in the cluster.

Source code in kmeanssa_ng/quantum_graph/lloyd_update.py 
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class MostFrequentNodeUpdate(LloydUpdateStrategy):
    """Update strategy that computes the new center as the most frequent
    closest node to the points in the cluster.
    """

    def __init__(self, random_state: int | np.random.Generator | None = None):
        # Normalize once so the tie-breaking stream advances across update()
        # calls instead of restarting from the same int seed each time.
        self._rng = (
            random_state
            if isinstance(random_state, np.random.Generator)
            else np.random.default_rng(random_state)
        )

    def update(self, points: list[QGPoint], space: "QuantumGraph") -> "QGCenter":
        """Compute the new center for a given cluster of points.

        Args:
            points: A list of points belonging to a single cluster.
            space: The quantum graph in which the points and center exist.

        Returns:
            The new center for the cluster.
        """
        if not points:
            # Or should this raise an error? Or return a random point?
            # For now, let's follow the old logic which would fail on empty.
            # Lloyd's should handle empty clusters.
            return None

        # Find the closest node for each point
        closest_nodes = []
        for p in points:
            if not isinstance(p, space.get_point_type()):
                raise TypeError("All points must be QGPoint instances.")
            closest_nodes.append(p.closest_node())

        # Count occurrences of each node
        node_counts = {}
        for node in closest_nodes:
            node_counts[node] = node_counts.get(node, 0) + 1

        # Find the most frequent node
        if not node_counts:
            raise ValueError("No valid nodes found for Fréchet mean approximation.")

        most_frequent_node = max(node_counts, key=node_counts.get)

        # Return a QGCenter at the most frequent node
        return space.node_as_center(most_frequent_node, rng=self._rng)

update(points, space)

Compute the new center for a given cluster of points.

Parameters:

Name Type Description Default
points list[QGPoint]

A list of points belonging to a single cluster.

 required
space 'QuantumGraph'

The quantum graph in which the points and center exist.

 required

Returns:

Type Description
'QGCenter'

The new center for the cluster.
Source code in kmeanssa_ng/quantum_graph/lloyd_update.py 
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def update(self, points: list[QGPoint], space: "QuantumGraph") -> "QGCenter":
    """Compute the new center for a given cluster of points.

    Args:
        points: A list of points belonging to a single cluster.
        space: The quantum graph in which the points and center exist.

    Returns:
        The new center for the cluster.
    """
    if not points:
        # Or should this raise an error? Or return a random point?
        # For now, let's follow the old logic which would fail on empty.
        # Lloyd's should handle empty clusters.
        return None

    # Find the closest node for each point
    closest_nodes = []
    for p in points:
        if not isinstance(p, space.get_point_type()):
            raise TypeError("All points must be QGPoint instances.")
        closest_nodes.append(p.closest_node())

    # Count occurrences of each node
    node_counts = {}
    for node in closest_nodes:
        node_counts[node] = node_counts.get(node, 0) + 1

    # Find the most frequent node
    if not node_counts:
        raise ValueError("No valid nodes found for Fréchet mean approximation.")

    most_frequent_node = max(node_counts, key=node_counts.get)

    # Return a QGCenter at the most frequent node
    return space.node_as_center(most_frequent_node, rng=self._rng)

QGCenter

Bases: QGPoint, Center

A movable cluster center on a quantum graph.

Centers can perform: - Brownian motion: Random walk for exploration - Drift: Directed movement toward target points

Attributes:

Name Type Description
space QuantumGraph

The quantum graph this center belongs to.
edge tuple[int, int]

The edge containing this center.
position tuple[int, int]

Position along the edge.
Example 
graph = QuantumGraph(...)
point = QGPoint(graph, edge=(0, 1), position=0.5)
center = QGCenter(point)
center.brownian_motion(0.1)
center.drift(target_point, 0.5)
Source code in kmeanssa_ng/quantum_graph/center.py 
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class QGCenter(QGPoint, Center):
    """A movable cluster center on a quantum graph.

    Centers can perform:
    - Brownian motion: Random walk for exploration
    - Drift: Directed movement toward target points

    Attributes:
        space: The quantum graph this center belongs to.
        edge: The edge containing this center.
        position: Position along the edge.

    Example:
        ```python
        graph = QuantumGraph(...)
        point = QGPoint(graph, edge=(0, 1), position=0.5)
        center = QGCenter(point)
        center.brownian_motion(0.1)
        center.drift(target_point, 0.5)
        ```
    """

    def __init__(
        self,
        point: QGPoint,
        rng: Generator | None = None,
    ) -> None:
        """Initialize a center from a point.

        Args:
            point: The initial point location.
            rng: Random number generator. If None, creates a new default_rng().
        """
        super().__init__(point.space, point.edge, point.position)
        self._rng = rng if rng is not None else default_rng()

    def _find_best_neighbor(self, n1: int, n2: int) -> int:
        """Find the neighbor of n1 whose edge starts a shortest walk to n2.

        The walk physically traverses the edge (n1, neighbor), so the cost of
        a candidate is the *length of that edge* plus the shortest path from
        the neighbor onward. Scoring with ``node_distance(n1, neighbor)``
        instead would tie a long direct edge with the true route whenever a
        shortcut exists (the shortest-path relaxation hides the edge length),
        and the drift would wander off the geodesic.

        Args:
            n1: Starting node.
            n2: Target node.

        Returns:
            A neighbor of n1 minimizing the walk to n2 (ties broken at random
            among neighbors that all realize the same minimal walk length).
        """
        if n1 == n2:
            return n1

        neighbors = list(self.space.neighbors(n1))
        min_distance = np.inf
        best_neighbors = []

        for neighbor in neighbors:
            distance = self.space.get_edge_length(n1, neighbor) + (
                self.space.node_distance(neighbor, n2)
            )
            if distance < min_distance:
                min_distance = distance
                best_neighbors = [neighbor]
            elif distance == min_distance:
                best_neighbors.append(neighbor)

        return best_neighbors[self._rng.integers(len(best_neighbors))]

    def drift(self, target_point: QGPoint, prop_to_travel: float) -> None:
        """Move toward a target point.

        Moves a proportion of the distance to the target point along
        the geodesic path in the quantum graph.

        Args:
            target_point: The point to move toward.
            prop_to_travel: Proportion of distance to travel (0 to 1).

        Raises:
            ValueError: If target_point is None, prop_to_travel is not numeric,
                or prop_to_travel is not in [0, 1].
        """
        # Validate target_point
        if target_point is None:
            raise ValueError("target_point cannot be None")

        # Validate prop_to_travel
        try:
            prop_float = float(prop_to_travel)
        except (TypeError, ValueError) as e:
            raise ValueError(
                f"prop_to_travel must be a number, got {type(prop_to_travel).__name__}"
            ) from e

        if prop_float < 0 or prop_float > 1:
            raise ValueError(f"prop_to_travel must be in [0, 1], got {prop_float}")

        quantum_path = self.space.quantum_path(self, target_point)
        path = quantum_path["path"]
        dist_to_target = quantum_path["distance"]
        dist_to_travel = prop_to_travel * dist_to_target

        # Case 1: Same edge (simple movement)
        if path is None:
            self._drift_on_same_edge(target_point, dist_to_travel)
        else:
            # Case 2: Different edges (traverse graph)
            self._drift_across_edges(target_point, path, dist_to_travel)

    def _drift_on_same_edge(self, target_point: QGPoint, dist_to_travel: float) -> None:
        """Handle drift when center and target are on the same physical edge."""
        edge_length = self.space.get_edge_length(*self.edge)

        if self.edge[0] == self.edge[1]:
            # Self-loop: move along the shorter arc, wrapping at the vertex.
            delta = target_point.position - self.position
            if delta > edge_length / 2:
                delta -= edge_length
            elif delta < -edge_length / 2:
                delta += edge_length
            step = dist_to_travel if delta >= 0 else -dist_to_travel
            self.position = (self.position + step) % edge_length
            return

        if self.edge == target_point.edge:
            target_position = target_point.position
        else:
            # Same physical edge, opposite parametrization: in this center's
            # frame the target sits at (edge length - target position).
            target_position = edge_length - target_point.position

        if self.position < target_position:
            self.position += dist_to_travel
        else:
            self.position -= dist_to_travel

    def _drift_across_edges(
        self,
        target_point: QGPoint,
        path: tuple[int, int],
        dist_to_travel: float,
    ) -> None:
        """Handle drift when center and target are on different edges."""
        next_node, target_node = path

        # Orient this center so the traversal exits forward (position
        # increasing). On a self-loop, reverse() is a reflection across the
        # vertex -- a *different physical point*, not a reparametrization --
        # so it is only safe when the walk provably leaves the loop (the
        # reflected and true trajectories then meet at the vertex). A walk
        # that stops on the loop must move along the true exit arc directly.
        if self.edge[0] == self.edge[1]:
            edge_length = self.space.get_edge_length(*self.edge)
            backward = self.position < edge_length - self.position
            arc = self.position if backward else edge_length - self.position
            if dist_to_travel < arc:
                self.position += -dist_to_travel if backward else dist_to_travel
                return
            if backward:
                self.reverse()
        elif self.edge[0] == next_node:
            self.reverse()

        # Orient the target locally: the observation belongs to the caller
        # and must never be mutated (on a self-loop, reverse() would even
        # move it to a different point of the graph).
        target_edge = target_point.edge
        target_length = self.space.get_edge_length(*target_edge)
        enter_far_arc = False
        if target_edge[0] == target_edge[1]:
            # Self-loop target: the geodesic enters via the shorter arc. Via
            # the far end, the coordinate runs backward from the loop length.
            enter_far_arc = (
                target_point.position > target_length - target_point.position
            )
        elif target_edge[1] == target_node:
            target_edge = (target_edge[1], target_edge[0])

        remaining_dist = dist_to_travel
        dist_to_next_node = self.space.get_edge_length(*self.edge) - self.position
        on_target_edge = False

        # Traverse edges until we've traveled the required distance
        while remaining_dist > dist_to_next_node:
            remaining_dist -= dist_to_next_node

            if target_node == next_node:
                # Reached target edge
                self.position = 0
                self.edge = target_edge
                on_target_edge = True
                dist_to_next_node = remaining_dist + 1  # Exit condition
            else:
                # Move to next edge
                self.position = 0
                cur_node = next_node
                next_node = self._find_best_neighbor(cur_node, target_node)
                self.edge = (cur_node, next_node)
                dist_to_next_node = self.space.get_edge_length(*self.edge)

        if on_target_edge and enter_far_arc:
            self.position = target_length - remaining_dist
        else:
            self.position += remaining_dist

    def clone(self) -> QGCenter:
        """Create an independent copy of this center.

        The cloned center shares the same quantum graph (space) but has
        independent edge and position attributes. This is much faster than
        deepcopy as it doesn't duplicate the entire graph structure.

        Note: This creates a shallow copy of the center's state, bypassing
        __init__ validation: the state comes from an existing center, so it
        is already valid, and cloning happens on the hot path of the
        annealing loop.

        Returns:
            A new QGCenter with the same location but independent state.

        Example:
            ```python
            original = QGCenter(...)
            copy = original.clone()
            original.brownian_motion(0.1)  # Doesn't affect copy
            ```
        """
        new_center = object.__new__(QGCenter)
        new_center._quantum_graph = self._quantum_graph
        new_center._edge = self._edge
        new_center.position = self.position
        new_center._rng = self._rng
        return new_center

    def brownian_motion(self, time_to_travel: float) -> None:
        """Perform Brownian motion on the quantum graph.

        The center performs a random walk with step size proportional
        to sqrt(time_to_travel).

        Args:
            time_to_travel: Time parameter (distance ~ sqrt(time)).

        Raises:
            ValueError: If time_to_travel is negative or not numeric.
        """
        # Validate time_to_travel
        try:
            time_float = float(time_to_travel)
        except (TypeError, ValueError) as e:
            raise ValueError(
                f"time_to_travel must be a number, got {type(time_to_travel).__name__}"
            ) from e

        if time_float < 0:
            raise ValueError(f"time_to_travel must be non-negative, got {time_float}")

        dist_to_travel = np.sqrt(time_float) * self._rng.standard_normal()
        edge_length = self.space.get_edge_length(*self.edge)

        forward = dist_to_travel > 0
        if forward:
            next_node = self.edge[1]
            dist_to_next_node = edge_length - self.position
        else:
            next_node = self.edge[0]
            dist_to_next_node = self.position

        remaining_dist = abs(dist_to_travel)

        # Traverse edges if motion exceeds current edge
        while remaining_dist >= dist_to_next_node:
            remaining_dist -= dist_to_next_node
            cur_node = next_node
            # Brownian motion on a metric graph leaves a vertex through an
            # *edge-end* chosen uniformly (Kirchhoff conditions). A self-loop
            # is attached to the vertex by both of its ends, so it counts
            # twice -- once entered forward (from position 0), once backward
            # (from the loop length).
            ends = []
            for nbr in self.space.neighbors(cur_node):
                ends.append((nbr, True))
                if nbr == cur_node:
                    ends.append((nbr, False))
            next_node, forward = ends[self._rng.integers(len(ends))]
            self.edge = (cur_node, next_node)
            edge_length = self.space.get_edge_length(*self.edge)
            # Whatever the sign of the initial draw, the leftover motion
            # continues away from the vertex: toward next_node on an edge
            # entered forward, toward decreasing positions on a self-loop
            # entered through its far end.
            self.position = 0 if forward else edge_length
            dist_to_next_node = edge_length

        # Final position update
        self.position += remaining_dist if forward else -remaining_dist

    def __repr__(self) -> str:
        """Detailed string representation."""
        return f"QGCenter(edge={self.edge}, position={self.position:.3f})"

    def __str__(self) -> str:
        """User-friendly string representation."""
        closest = self.closest_node()
        return f"Center near node {closest} [edge {self.edge}, pos={self.position:.3f}]"

__init__(point, rng=None)

Initialize a center from a point.

Parameters:

Name Type Description Default
point QGPoint

The initial point location.

 required
rng Generator | None

Random number generator. If None, creates a new default_rng().

 None
Source code in kmeanssa_ng/quantum_graph/center.py 
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def __init__(
    self,
    point: QGPoint,
    rng: Generator | None = None,
) -> None:
    """Initialize a center from a point.

    Args:
        point: The initial point location.
        rng: Random number generator. If None, creates a new default_rng().
    """
    super().__init__(point.space, point.edge, point.position)
    self._rng = rng if rng is not None else default_rng()

__repr__()

Detailed string representation.

Source code in kmeanssa_ng/quantum_graph/center.py 
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def __repr__(self) -> str:
    """Detailed string representation."""
    return f"QGCenter(edge={self.edge}, position={self.position:.3f})"

__str__()

User-friendly string representation.

Source code in kmeanssa_ng/quantum_graph/center.py 
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def __str__(self) -> str:
    """User-friendly string representation."""
    closest = self.closest_node()
    return f"Center near node {closest} [edge {self.edge}, pos={self.position:.3f}]"

brownian_motion(time_to_travel)

Perform Brownian motion on the quantum graph.

The center performs a random walk with step size proportional to sqrt(time_to_travel).

Parameters:

Name Type Description Default
time_to_travel float

Time parameter (distance ~ sqrt(time)).

 required

Raises:

Type Description
ValueError

If time_to_travel is negative or not numeric.
Source code in kmeanssa_ng/quantum_graph/center.py 
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def brownian_motion(self, time_to_travel: float) -> None:
    """Perform Brownian motion on the quantum graph.

    The center performs a random walk with step size proportional
    to sqrt(time_to_travel).

    Args:
        time_to_travel: Time parameter (distance ~ sqrt(time)).

    Raises:
        ValueError: If time_to_travel is negative or not numeric.
    """
    # Validate time_to_travel
    try:
        time_float = float(time_to_travel)
    except (TypeError, ValueError) as e:
        raise ValueError(
            f"time_to_travel must be a number, got {type(time_to_travel).__name__}"
        ) from e

    if time_float < 0:
        raise ValueError(f"time_to_travel must be non-negative, got {time_float}")

    dist_to_travel = np.sqrt(time_float) * self._rng.standard_normal()
    edge_length = self.space.get_edge_length(*self.edge)

    forward = dist_to_travel > 0
    if forward:
        next_node = self.edge[1]
        dist_to_next_node = edge_length - self.position
    else:
        next_node = self.edge[0]
        dist_to_next_node = self.position

    remaining_dist = abs(dist_to_travel)

    # Traverse edges if motion exceeds current edge
    while remaining_dist >= dist_to_next_node:
        remaining_dist -= dist_to_next_node
        cur_node = next_node
        # Brownian motion on a metric graph leaves a vertex through an
        # *edge-end* chosen uniformly (Kirchhoff conditions). A self-loop
        # is attached to the vertex by both of its ends, so it counts
        # twice -- once entered forward (from position 0), once backward
        # (from the loop length).
        ends = []
        for nbr in self.space.neighbors(cur_node):
            ends.append((nbr, True))
            if nbr == cur_node:
                ends.append((nbr, False))
        next_node, forward = ends[self._rng.integers(len(ends))]
        self.edge = (cur_node, next_node)
        edge_length = self.space.get_edge_length(*self.edge)
        # Whatever the sign of the initial draw, the leftover motion
        # continues away from the vertex: toward next_node on an edge
        # entered forward, toward decreasing positions on a self-loop
        # entered through its far end.
        self.position = 0 if forward else edge_length
        dist_to_next_node = edge_length

    # Final position update
    self.position += remaining_dist if forward else -remaining_dist

clone()

Create an independent copy of this center.

The cloned center shares the same quantum graph (space) but has independent edge and position attributes. This is much faster than deepcopy as it doesn't duplicate the entire graph structure.

Note: This creates a shallow copy of the center's state, bypassing init validation: the state comes from an existing center, so it is already valid, and cloning happens on the hot path of the annealing loop.

Returns:

Type Description
QGCenter

A new QGCenter with the same location but independent state.
Example 
original = QGCenter(...)
copy = original.clone()
original.brownian_motion(0.1)  # Doesn't affect copy
Source code in kmeanssa_ng/quantum_graph/center.py 
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def clone(self) -> QGCenter:
    """Create an independent copy of this center.

    The cloned center shares the same quantum graph (space) but has
    independent edge and position attributes. This is much faster than
    deepcopy as it doesn't duplicate the entire graph structure.

    Note: This creates a shallow copy of the center's state, bypassing
    __init__ validation: the state comes from an existing center, so it
    is already valid, and cloning happens on the hot path of the
    annealing loop.

    Returns:
        A new QGCenter with the same location but independent state.

    Example:
        ```python
        original = QGCenter(...)
        copy = original.clone()
        original.brownian_motion(0.1)  # Doesn't affect copy
        ```
    """
    new_center = object.__new__(QGCenter)
    new_center._quantum_graph = self._quantum_graph
    new_center._edge = self._edge
    new_center.position = self.position
    new_center._rng = self._rng
    return new_center

drift(target_point, prop_to_travel)

Move toward a target point.

Moves a proportion of the distance to the target point along the geodesic path in the quantum graph.

Parameters:

Name Type Description Default
target_point QGPoint

The point to move toward.

 required
prop_to_travel float

Proportion of distance to travel (0 to 1).

 required

Raises:

Type Description
ValueError

If target_point is None, prop_to_travel is not numeric, or prop_to_travel is not in [0, 1].
Source code in kmeanssa_ng/quantum_graph/center.py 
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def drift(self, target_point: QGPoint, prop_to_travel: float) -> None:
    """Move toward a target point.

    Moves a proportion of the distance to the target point along
    the geodesic path in the quantum graph.

    Args:
        target_point: The point to move toward.
        prop_to_travel: Proportion of distance to travel (0 to 1).

    Raises:
        ValueError: If target_point is None, prop_to_travel is not numeric,
            or prop_to_travel is not in [0, 1].
    """
    # Validate target_point
    if target_point is None:
        raise ValueError("target_point cannot be None")

    # Validate prop_to_travel
    try:
        prop_float = float(prop_to_travel)
    except (TypeError, ValueError) as e:
        raise ValueError(
            f"prop_to_travel must be a number, got {type(prop_to_travel).__name__}"
        ) from e

    if prop_float < 0 or prop_float > 1:
        raise ValueError(f"prop_to_travel must be in [0, 1], got {prop_float}")

    quantum_path = self.space.quantum_path(self, target_point)
    path = quantum_path["path"]
    dist_to_target = quantum_path["distance"]
    dist_to_travel = prop_to_travel * dist_to_target

    # Case 1: Same edge (simple movement)
    if path is None:
        self._drift_on_same_edge(target_point, dist_to_travel)
    else:
        # Case 2: Different edges (traverse graph)
        self._drift_across_edges(target_point, path, dist_to_travel)

QGPoint

Bases: Point

A point on a quantum graph.

A quantum graph point is located on an edge at a specific position. The edge is represented as a tuple (node1, node2) and the position is the distance from node1 along the edge.

Attributes:

Name Type Description
space QuantumGraph

The quantum graph this point belongs to.
edge tuple[int, int]

The edge (node1, node2) containing this point.
position tuple[int, int]

Distance from node1 along the edge.
Example 
graph = QuantumGraph(...)
point = QGPoint(graph, edge=(0, 1), position=0.5)
Source code in kmeanssa_ng/quantum_graph/point.py 
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class QGPoint(Point):
    """A point on a quantum graph.

    A quantum graph point is located on an edge at a specific position.
    The edge is represented as a tuple (node1, node2) and the position
    is the distance from node1 along the edge.

    Attributes:
        space: The quantum graph this point belongs to.
        edge: The edge (node1, node2) containing this point.
        position: Distance from node1 along the edge.

    Example:
        ```python
        graph = QuantumGraph(...)
        point = QGPoint(graph, edge=(0, 1), position=0.5)
        ```
    """

    def __init__(
        self,
        quantum_graph: QuantumGraph,
        edge: tuple[int, int],
        position: float,
    ) -> None:
        """Initialize a point on a quantum graph.

        Args:
            quantum_graph: The quantum graph containing this point.
            edge: Tuple (node1, node2) representing the edge.
            position: Distance from node1 along the edge.

        Raises:
            ValueError: If quantum_graph is None, edge doesn't exist in graph,
                or position is outside [0, edge_length].
        """
        if quantum_graph is None:
            raise ValueError("quantum_graph cannot be None")

        if not isinstance(edge, tuple) or len(edge) != 2:
            raise ValueError(f"edge must be a tuple of two nodes, got {edge}")

        # Check edge exists in graph (allow self-loops)
        if edge[0] != edge[1] and edge not in quantum_graph.edges:
            # Try reversed edge
            if (edge[1], edge[0]) not in quantum_graph.edges:
                raise ValueError(f"Edge {edge} does not exist in the graph")

        self._quantum_graph = quantum_graph
        self._edge = edge

        # Validate and set position
        self._validate_and_set_position(position)

    def _validate_and_set_position(self, position: float) -> None:
        """Validate and set the position on the edge.

        Args:
            position: Position along the edge from node1.

        Raises:
            ValueError: If position is not numeric or outside [0, edge_length].
        """
        # Check if position is numeric
        try:
            position_float = float(position)
        except (TypeError, ValueError) as e:
            raise ValueError(
                f"Position must be a number, got {type(position).__name__}"
            ) from e

        # Check if position is non-negative
        if position_float < 0:
            raise ValueError(f"Position must be non-negative, got {position_float}")

        # Check if position is within edge length
        edge_length = self.space.get_edge_length(*self._edge)
        if position_float > edge_length:
            raise ValueError(
                f"Position {position_float} exceeds edge length {edge_length} for edge {self._edge}"
            )

        self.position = position_float

    @property
    def space(self) -> QuantumGraph:
        """The quantum graph this point belongs to."""
        return self._quantum_graph

    @property
    def edge(self) -> tuple[int, int]:
        """The edge containing this point."""
        if self._edge not in self.space.edges:
            if self._edge[0] == self._edge[1]:
                return self._edge
            else:
                raise ValueError("The edge does not belong to the graph")
        return self._edge

    @edge.setter
    def edge(self, new_edge: tuple[int, int]) -> None:
        """Set the edge containing this point.

        Args:
            new_edge: New edge (node1, node2).

        Raises:
            ValueError: If the edge doesn't belong to the graph.

        Note:
            Position validation is not performed here to allow internal
            operations (brownian_motion, drift) to change edge first, then
            update position. Position validation is done in __init__.
        """
        if not ((new_edge in self.space.edges) or (new_edge[0] == new_edge[1])):
            raise ValueError("The edge does not belong to the graph")

        self._edge = new_edge

    def closest_node(self) -> int:
        """Get the closest node to this point.

        Returns:
            The node (edge[0] or edge[1]) closest to this point.
        """
        edge_length = self.space.get_edge_length(*self.edge)
        if self.position < edge_length / 2:
            return self.edge[0]
        else:
            return self.edge[1]

    def reverse(self) -> None:
        """Reverse the edge orientation and adjust position.

        Changes edge from (a, b) to (b, a) and updates position accordingly.
        """
        self.edge = (self.edge[1], self.edge[0])
        edge_length = self.space.get_edge_length(*self.edge)
        self.position = edge_length - self.position

    def __str__(self) -> str:
        """String representation of the point."""
        name = (
            f" '{self.space.name}'"
            if hasattr(self.space, "name") and self.space.name
            else ""
        )
        return f"QGPoint on{name} edge {self.edge} at position {self.position:.3f}"

    def __repr__(self) -> str:
        """Detailed string representation."""
        return f"QGPoint(edge={self.edge}, position={self.position})"

edge property writable

The edge containing this point.

space property

The quantum graph this point belongs to.

__init__(quantum_graph, edge, position)

Initialize a point on a quantum graph.

Parameters:

Name Type Description Default
quantum_graph QuantumGraph

The quantum graph containing this point.

 required
edge tuple[int, int]

Tuple (node1, node2) representing the edge.

 required
position float

Distance from node1 along the edge.

 required

Raises:

Type Description
ValueError

If quantum_graph is None, edge doesn't exist in graph, or position is outside [0, edge_length].
Source code in kmeanssa_ng/quantum_graph/point.py 
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def __init__(
    self,
    quantum_graph: QuantumGraph,
    edge: tuple[int, int],
    position: float,
) -> None:
    """Initialize a point on a quantum graph.

    Args:
        quantum_graph: The quantum graph containing this point.
        edge: Tuple (node1, node2) representing the edge.
        position: Distance from node1 along the edge.

    Raises:
        ValueError: If quantum_graph is None, edge doesn't exist in graph,
            or position is outside [0, edge_length].
    """
    if quantum_graph is None:
        raise ValueError("quantum_graph cannot be None")

    if not isinstance(edge, tuple) or len(edge) != 2:
        raise ValueError(f"edge must be a tuple of two nodes, got {edge}")

    # Check edge exists in graph (allow self-loops)
    if edge[0] != edge[1] and edge not in quantum_graph.edges:
        # Try reversed edge
        if (edge[1], edge[0]) not in quantum_graph.edges:
            raise ValueError(f"Edge {edge} does not exist in the graph")

    self._quantum_graph = quantum_graph
    self._edge = edge

    # Validate and set position
    self._validate_and_set_position(position)

__repr__()

Detailed string representation.

Source code in kmeanssa_ng/quantum_graph/point.py 
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def __repr__(self) -> str:
    """Detailed string representation."""
    return f"QGPoint(edge={self.edge}, position={self.position})"

__str__()

String representation of the point.

Source code in kmeanssa_ng/quantum_graph/point.py 
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def __str__(self) -> str:
    """String representation of the point."""
    name = (
        f" '{self.space.name}'"
        if hasattr(self.space, "name") and self.space.name
        else ""
    )
    return f"QGPoint on{name} edge {self.edge} at position {self.position:.3f}"

closest_node()

Get the closest node to this point.

Returns:

Type Description
int

The node (edge[0] or edge[1]) closest to this point.
Source code in kmeanssa_ng/quantum_graph/point.py 
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def closest_node(self) -> int:
    """Get the closest node to this point.

    Returns:
        The node (edge[0] or edge[1]) closest to this point.
    """
    edge_length = self.space.get_edge_length(*self.edge)
    if self.position < edge_length / 2:
        return self.edge[0]
    else:
        return self.edge[1]

reverse()

Reverse the edge orientation and adjust position.

Changes edge from (a, b) to (b, a) and updates position accordingly.

Source code in kmeanssa_ng/quantum_graph/point.py 
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def reverse(self) -> None:
    """Reverse the edge orientation and adjust position.

    Changes edge from (a, b) to (b, a) and updates position accordingly.
    """
    self.edge = (self.edge[1], self.edge[0])
    edge_length = self.space.get_edge_length(*self.edge)
    self.position = edge_length - self.position

QuantumGraph

Bases: Graph, Space

A quantum graph is a metric graph where points can lie on edges.

This class extends NetworkX Graph to provide: - Distance computation between points on edges - Sampling of random points and centers - k-means clustering support

Each edge should have a 'length' attribute representing its metric length. Nodes and edges can have 'weight' and 'distribution' attributes for sampling.

Attributes:

Name Type Description
diameter float

The diameter of the graph (max distance between nodes).
node_position dict

Layout positions for visualization.
Example 
graph = QuantumGraph()
graph.add_edge(0, 1, length=1.0)
graph.add_edge(1, 2, length=2.0)
graph.precomputing()  # Cache pairwise distances

points = graph.sample_points(100)
centers = graph.sample_centers(5)
Source code in kmeanssa_ng/quantum_graph/space.py 
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class QuantumGraph(nx.Graph, Space):
    """A quantum graph is a metric graph where points can lie on edges.

    This class extends NetworkX Graph to provide:
    - Distance computation between points on edges
    - Sampling of random points and centers
    - k-means clustering support

    Each edge should have a 'length' attribute representing its metric length.
    Nodes and edges can have 'weight' and 'distribution' attributes for sampling.

    Attributes:
        diameter: The diameter of the graph (max distance between nodes).
        node_position: Layout positions for visualization.

    Example:
        ```python
        graph = QuantumGraph()
        graph.add_edge(0, 1, length=1.0)
        graph.add_edge(1, 2, length=2.0)
        graph.precomputing()  # Cache pairwise distances

        points = graph.sample_points(100)
        centers = graph.sample_centers(5)
        ```
    """

    def __init__(
        self, incoming_graph_data=None, precompute: bool = False, **attr
    ) -> None:
        """Initialize a quantum graph.

        Args:
            incoming_graph_data: Input graph data (see networkx.Graph).
            precompute: If True, automatically precompute distances after initialization.
            **attr: Additional graph attributes.
        """
        super().__init__(incoming_graph_data, **attr)
        self._pairwise_nodes_distance: dict[int, dict[int, float]] | None = None
        self._pairwise_nodes_distance_array: np.ndarray | None = None
        self._node_to_index: dict[int, int] | None = None
        self._diameter: float = 0.0
        self._node_position: dict | None = None

        # Validate any length carried by the incoming data, so a degenerate
        # edge (e.g. a zero-length edge from a collapsed epsilon-net) fails
        # here with a clear ValueError rather than silently poisoning a later
        # precomputing() when precompute=False. Only lengths that are present
        # are checked, so building the topology first and setting the lengths
        # afterwards (generate_sbm, as_quantum_graph) still works.
        for u, v, length in self.edges(data="length"):
            if length is not None:
                self._validate_length(u, v, length)

        if precompute and self.number_of_nodes() > 0:
            self.precomputing()

    def _invalidate_distance_cache(self) -> None:
        """Drop every cache derived from the graph structure.

        Called by the graph mutators so that stale distances are never served
        after an edit: distance queries then fail loudly (or recompute) until
        ``precomputing()`` is run again.
        """
        self._pairwise_nodes_distance = None
        self._pairwise_nodes_distance_array = None
        self._node_to_index = None
        self._diameter = 0.0
        self._node_position = None

    def add_edge(self, u_for_edge, v_for_edge, **attr) -> None:
        """Add an edge with validation of the length attribute.

        Invalidates the precomputed distance cache: call ``precomputing()``
        again after editing the graph.

        Args:
            u_for_edge: First node.
            v_for_edge: Second node.
            **attr: Edge attributes. Must include 'length' with a positive value.

        Raises:
            ValueError: If 'length' is missing, not positive, or not a number.
        """
        if "length" not in attr:
            raise ValueError(
                f"Edge ({u_for_edge}, {v_for_edge}) must have a 'length' attribute"
            )

        self._validate_length(u_for_edge, v_for_edge, attr["length"])

        super().add_edge(u_for_edge, v_for_edge, **attr)
        self._invalidate_distance_cache()

    @staticmethod
    def _validate_length(u, v, length) -> None:
        """Validate a single edge length (a positive number)."""
        try:
            length_float = float(length)
        except (TypeError, ValueError) as e:
            raise ValueError(
                f"Edge ({u}, {v}) length must be a number, got {type(length).__name__}"
            ) from e
        if length_float <= 0:
            raise ValueError(
                f"Edge ({u}, {v}) length must be positive, got {length_float}"
            )

    def add_node(self, node_for_adding, **attr) -> None:
        """Add a node; invalidates the precomputed distance cache."""
        super().add_node(node_for_adding, **attr)
        self._invalidate_distance_cache()

    def remove_edge(self, u, v) -> None:
        """Remove an edge; invalidates the precomputed distance cache."""
        super().remove_edge(u, v)
        self._invalidate_distance_cache()

    def remove_node(self, n) -> None:
        """Remove a node; invalidates the precomputed distance cache."""
        super().remove_node(n)
        self._invalidate_distance_cache()

    # The bulk mutators inherited from networkx.Graph do not route through the
    # add_edge/add_node overrides above, so without these the distance cache
    # would be served stale after a bulk edit (and the graph constructor, which
    # feeds incoming data through add_edges_from/add_nodes_from, would bypass
    # cache invalidation entirely). Each override invalidates the cache once for
    # the whole batch. Length validation is not done here: the constructor
    # routes incoming data through add_edges_from, and networkx wraps any
    # exception raised there into an opaque NetworkXError; supplied lengths are
    # validated once in __init__ instead (a clean ValueError), which also keeps
    # the "add the topology, then set the lengths" pattern working.

    def add_edges_from(self, ebunch_to_add, **attr) -> None:
        """Add edges in bulk; invalidates the precomputed distance cache."""
        super().add_edges_from(ebunch_to_add, **attr)
        self._invalidate_distance_cache()

    def add_nodes_from(self, nodes_for_adding, **attr) -> None:
        """Add nodes in bulk; invalidates the precomputed distance cache."""
        super().add_nodes_from(nodes_for_adding, **attr)
        self._invalidate_distance_cache()

    def remove_edges_from(self, ebunch) -> None:
        """Remove edges in bulk; invalidates the precomputed distance cache."""
        super().remove_edges_from(ebunch)
        self._invalidate_distance_cache()

    def remove_nodes_from(self, nodes) -> None:
        """Remove nodes in bulk; invalidates the precomputed distance cache."""
        super().remove_nodes_from(nodes)
        self._invalidate_distance_cache()

    def clear(self) -> None:
        """Clear the graph; invalidates the precomputed distance cache."""
        super().clear()
        self._invalidate_distance_cache()

    def clear_edges(self) -> None:
        """Clear all edges; invalidates the precomputed distance cache."""
        super().clear_edges()
        self._invalidate_distance_cache()

    @property
    def diameter(self) -> float:
        """Compute and cache the graph diameter.

        Returns:
            Maximum distance between any two nodes.
        """
        if self._diameter == 0.0:
            for n1 in self.nodes:
                for n2 in self.nodes:
                    d = self.node_distance(n1, n2)
                    if d > self._diameter:
                        self._diameter = d
        return self._diameter

    def validate_edge_lengths(self) -> None:
        """Validate that all edges have positive length attributes.

        Raises:
            ValueError: If any edge is missing 'length' or has invalid length.
        """
        for u, v, data in self.edges(data=True):
            if "length" not in data:
                raise ValueError(f"Edge ({u}, {v}) missing 'length' attribute")

            length = data["length"]
            try:
                length_float = float(length)
            except (TypeError, ValueError) as e:
                raise ValueError(
                    f"Edge ({u}, {v}) length must be a number, got {type(length).__name__}"
                ) from e

            if length_float <= 0:
                raise ValueError(
                    f"Edge ({u}, {v}) has invalid length {length_float}, must be positive"
                )

    def precomputing(self) -> None:
        """Precompute and cache all pairwise node distances.

        This significantly speeds up distance queries.
        Should be called once after graph construction.

        Raises:
            ValueError: If graph is not connected or has invalid edge lengths.
        """
        # Validate edge lengths first
        self.validate_edge_lengths()

        # Check connectivity
        if self.number_of_nodes() > 0 and not nx.is_connected(self):
            num_components = nx.number_connected_components(self)
            raise ValueError(
                f"Graph must be connected for distance precomputing. "
                f"Found {num_components} connected components."
            )

        if self._pairwise_nodes_distance is None:
            self._pairwise_nodes_distance = dict(
                nx.all_pairs_dijkstra_path_length(self, weight="length")
            )

            # Create numpy array version for Numba-accelerated computations
            node_list = list(self.nodes())
            n = len(node_list)
            self._node_to_index = {node: i for i, node in enumerate(node_list)}
            self._pairwise_nodes_distance_array = np.zeros((n, n), dtype=np.float64)

            for i, node_i in enumerate(node_list):
                for j, node_j in enumerate(node_list):
                    self._pairwise_nodes_distance_array[i, j] = (
                        self._pairwise_nodes_distance[node_i][node_j]
                    )

    @property
    def node_position(self) -> dict:
        """Compute layout positions for visualization.

        Returns:
            Dictionary mapping nodes to (x, y) positions.
        """
        if self._node_position is None:
            drawing = nx.get_edge_attributes(self, "length")
            drawing_weights = {
                (i, j): {"drawing": 1 / drawing[(i, j)]} for i, j in drawing
            }
            nx.set_edge_attributes(self, drawing_weights)
            # Fixed seed: the layout is the force-directed embedding of the
            # graph, only used for drawing, but leaving it unseeded made every
            # figure non-reproducible, breaking the package's determinism
            # guarantee.
            self._node_position = nx.layout.spring_layout(
                self, weight="drawing", seed=0
            )
        return self._node_position

    def node_distance(self, n1: int, n2: int) -> float:
        """Compute shortest path distance between two nodes.

        Args:
            n1: First node.
            n2: Second node.

        Returns:
            Shortest path length using 'length' edge attribute.
        """
        # Prefer the precomputed array (O(1)); it may be present without the dict
        # form (e.g. loaded from a cache). Fall back to the dict, then to a live
        # Dijkstra when nothing is precomputed.
        if self._pairwise_nodes_distance_array is not None and self._node_to_index:
            return float(
                self._pairwise_nodes_distance_array[
                    self._node_to_index[n1], self._node_to_index[n2]
                ]
            )
        if self._pairwise_nodes_distance is not None:
            return self._pairwise_nodes_distance[n1][n2]
        return nx.shortest_path_length(self, n1, n2, weight="length")

    def get_edge_length(self, n1: int, n2: int) -> float:
        """Get the length of an edge.

        Args:
            n1: First node of the edge.
            n2: Second node of the edge.

        Returns:
            Edge length.

        Raises:
            ValueError: If edge does not exist.
        """
        edge_data = self.get_edge_data(n1, n2)
        if edge_data is None:
            raise ValueError(f"Edge ({n1}, {n2}) does not exist in graph")
        return edge_data["length"]

    def quantum_path(self, p1: QGPoint, p2: QGPoint) -> dict[str, float | tuple | None]:
        """Compute the geodesic between two points on the graph.

        Args:
            p1: First point.
            p2: Second point.

        Returns:
            Dictionary with:
                - 'distance': geodesic distance
                - 'path': (node_from_p1_edge, node_from_p2_edge) or None if same edge
        """
        edge1, edge2 = p1.edge, p2.edge
        pos1, pos2 = p1.position, p2.position
        length1 = self.get_edge_length(*edge1)
        length2 = self.get_edge_length(*edge2)

        # Both points on the same self-loop: the geodesic stays on the loop
        # (direct arc or the complementary one, possibly through the vertex).
        if edge1[0] == edge1[1] and edge2[0] == edge2[1] and edge1[0] == edge2[0]:
            arc = abs(pos1 - pos2)
            return {"distance": min(arc, length1 - arc), "path": None}

        # Compute distances for all 4 possible paths
        d0 = self.node_distance(edge1[0], edge2[0]) + pos1 + pos2
        d1 = self.node_distance(edge1[0], edge2[1]) + pos1 + (length2 - pos2)
        d2 = self.node_distance(edge1[1], edge2[0]) + (length1 - pos1) + pos2
        d3 = (
            self.node_distance(edge1[1], edge2[1]) + (length1 - pos1) + (length2 - pos2)
        )

        # Find minimum distance (break ties deterministically)
        distances = np.array([d0, d1, d2, d3])
        all_idx = np.where(distances == distances.min())[0]
        idx = all_idx[0]
        d_min = distances[idx]

        # Check if points are on the same (non-loop) physical edge
        if (
            edge1[0] != edge1[1]
            and edge1[0] == edge2[1]
            and edge1[1] == edge2[0]
            and d_min > abs(length1 - pos1 - pos2)
        ):
            return {"distance": abs(length1 - pos1 - pos2), "path": None}

        if edge1[0] != edge1[1] and edge1 == edge2 and abs(pos1 - pos2) < d_min:
            return {"distance": abs(pos1 - pos2), "path": None}

        # Determine path nodes
        if idx == 0:
            path = (edge1[0], edge2[0])
        elif idx == 1:
            path = (edge1[0], edge2[1])
        elif idx == 2:
            path = (edge1[1], edge2[0])
        else:
            path = (edge1[1], edge2[1])

        return {"distance": d_min, "path": path}

    def distance(self, p1: QGPoint, p2: QGPoint) -> float:
        """Compute distance between two points on the graph.

        Args:
            p1: First point.
            p2: Second point.

        Returns:
            Geodesic distance.
        """
        return self.quantum_path(p1, p2)["distance"]

    def distances_from_centers(
        self, centers: list[QGCenter], target: QGPoint
    ) -> np.ndarray:
        """Compute distances from multiple centers to a single target point.

        This is a Numba-accelerated operation that efficiently computes distances
        from all centers to one target point. Works for any target location
        (on nodes or edges).

        Args:
            centers: List of k centers to compute distances from.
            target: The target point (can be on a node or edge).

        Returns:
            Array of shape (k,) with distances from each center to target.

        Raises:
            ValueError: If pairwise distances not precomputed.

        Example:
            ```python
            centers = graph.sample_centers(5)
            target = graph.sample_points(1)[0]
            distances = graph.distances_from_centers(centers, target)
            closest_idx = np.argmin(distances)
            closest_center = centers[closest_idx]
            ```
        """
        if self._pairwise_nodes_distance_array is None:
            raise ValueError("Must call precomputing() before distances_from_centers")

        k = len(centers)

        # Extract center data into numpy arrays for Numba
        center_edges_0 = np.empty(k, dtype=np.int32)
        center_edges_1 = np.empty(k, dtype=np.int32)
        center_positions = np.empty(k, dtype=np.float64)
        center_lengths = np.empty(k, dtype=np.float64)

        for i, center in enumerate(centers):
            edge = center.edge
            center_edges_0[i] = self._node_to_index[edge[0]]
            center_edges_1[i] = self._node_to_index[edge[1]]
            center_positions[i] = center.position
            center_lengths[i] = self.get_edge_length(*edge)

        # Extract target information
        target_edge = target.edge
        target_edge_0 = self._node_to_index[target_edge[0]]
        target_edge_1 = self._node_to_index[target_edge[1]]
        target_pos = target.position
        target_length = self.get_edge_length(*target_edge)

        # Call Numba-accelerated function
        return _batch_distances_numba(
            center_edges_0,
            center_edges_1,
            center_positions,
            center_lengths,
            target_edge_0,
            target_edge_1,
            target_pos,
            target_length,
            self._pairwise_nodes_distance_array,
        )

    def node_center_distances(self, centers: list[QGCenter]) -> np.ndarray:
        """Geodesic distance from every node to every center.

        Args:
            centers: List of k centers.

        Returns:
            Array of shape (n_nodes, k), with rows in ``list(self.nodes())``
            order: entry (v, i) is the geodesic distance from node v to center i.

        Raises:
            ValueError: If pairwise distances have not been precomputed.
        """
        if self._pairwise_nodes_distance_array is None:
            raise ValueError("Must call precomputing() before node_center_distances")

        distances = self._pairwise_nodes_distance_array  # (n, n), node order
        index = self._node_to_index
        out = np.empty((distances.shape[0], len(centers)))
        for i, center in enumerate(centers):
            a, b = center.edge
            length = self.get_edge_length(a, b)
            # A center sits at `position` along edge (a, b); a node reaches it
            # via whichever endpoint is closer.
            out[:, i] = np.minimum(
                center.position + distances[:, index[a]],
                (length - center.position) + distances[:, index[b]],
            )
        return out

    def node_energy(
        self, centers: list[QGCenter], weights: np.ndarray | None = None
    ) -> float:
        """Node-weighted k-means energy ``sum_v w[v] * min_i d(v, c_i)^2``.

        Args:
            centers: List of cluster centers.
            weights: Per-node weights in ``list(self.nodes())`` order. Defaults
                to the nodes' ``weight`` attribute (1.0 when absent).

        Returns:
            The weighted energy.
        """
        min_sq = self.node_center_distances(centers).min(axis=1) ** 2
        if weights is None:
            weights = np.array(
                [self.nodes[node].get("weight", 1.0) for node in self.nodes()]
            )
        return float(np.sum(np.asarray(weights) * min_sq))

    def center_from_point(self, point: QGPoint, rng=None) -> QGCenter:
        """Create a QGCenter object from a QGPoint object."""
        return QGCenter(point, rng=rng)

    def nodes_as_points(self) -> list[QGPoint]:
        """Convert all nodes to points.

        Returns:
            List of points, one at each node.
        """
        points = []
        for node in self.nodes:
            neighbor = next(self.neighbors(node))
            points.append(QGPoint(self, (node, neighbor), 0))
        return points

    def node_as_center(self, node: int, rng=None) -> QGCenter:
        """Create a center at a specific node.

        Args:
            node: The node to place the center at.
            rng: Optional random number generator for neighbor selection.

        Returns:
            A center located at the node.
        """
        if rng is not None:
            nbrs = list(self.neighbors(node))
            neighbor = nbrs[rng.integers(len(nbrs))]
        else:
            neighbor = next(self.neighbors(node))
        point = QGPoint(self, (node, neighbor), 0)
        return QGCenter(point, rng=rng)

    def calculate_energy(
        self,
        centers: list[QGCenter],
        how: str = "uniform",
        observations: list[QGPoint] | None = None,
    ) -> float:
        """Calculate k-means energy for given centers.

        Dispatches to the Numba kernels when pairwise node distances are
        precomputed (``precomputing()``), and falls back to pure-Python
        distance computations otherwise — correct on any graph, just slower.

        Args:
            centers: List of cluster centers.
            how: Which reference measure to average under (see
                :meth:`Space.calculate_energy` for the full contract):
                - "uniform": average over nodes with equal weight; rejects
                  ``observations``.
                - "empirical": average over the ``observations`` list
                  (required), each point exactly where it lies on its edge —
                  no rounding to nodes.
                - "node_measure": average over nodes weighted by their
                  ``obs_weight``, set explicitly by the caller (e.g. through
                  ``register_observations``); rejects ``observations``.
            observations: The points defining the empirical measure
                (``how="empirical"`` only).

        Returns:
            Average squared distance to nearest center.

        Raises:
            ValueError: If ``how`` is unknown (including the retired
                ``"obs"``), if ``"empirical"`` lacks observations, if
                ``"uniform"``/``"node_measure"`` receive some, or if no node
                carries a positive ``obs_weight`` in ``"node_measure"`` mode
                (the energy would silently be 0.0 for every configuration
                of centers).
        """
        if how == "obs":
            raise ValueError(
                "energy mode 'obs' was split into two explicit modes: use "
                "'empirical' to average over an explicit list of observation "
                "points, or 'node_measure' to average under the per-node "
                "'obs_weight' measure registered on the graph."
            )
        if how not in ("uniform", "empirical", "node_measure"):
            raise ValueError(
                f"how must be 'uniform', 'empirical' or 'node_measure', got {how!r}"
            )
        if how == "empirical":
            if not observations:
                raise ValueError(
                    "energy mode 'empirical' requires the explicit "
                    "'observations' list defining the empirical measure"
                )
        elif observations is not None:
            raise ValueError(
                f"energy mode {how!r} defines its own reference measure and "
                "does not take 'observations'; pass them with how='empirical' "
                "instead"
            )
        if how == "node_measure" and not any(
            data.get("obs_weight", 0) > 0 for _, data in self.nodes(data=True)
        ):
            raise ValueError(
                "energy mode 'node_measure' requires a measure on the graph "
                "nodes, but no node carries a positive 'obs_weight'. Register "
                "one explicitly: register_observations(points) counts each "
                "point at its closest node, or set the 'obs_weight' node "
                "attribute directly (e.g. to a population measure)."
            )
        if self._pairwise_nodes_distance_array is not None:
            return self.calculate_energy_numba(
                centers, how=how, observations=observations
            )
        return self._calculate_energy_python(centers, how, observations=observations)

    def register_observations(self, points: list[QGPoint]) -> None:
        """Set the per-node observation measure ``obs_weight`` from these points.

        Each point counts at its closest node. This **replaces** any previous
        measure — the measure describes one observation set — so callers
        combining several samples must register the union explicitly.

        Args:
            points: The observation points (any points of this graph).
        """
        nx.set_node_attributes(self, 0, "obs_weight")
        counts: dict = {}
        for point in points:
            node = point.closest_node()
            counts[node] = counts.get(node, 0) + 1
        nx.set_node_attributes(self, counts, "obs_weight")

    def _calculate_energy_python(
        self,
        centers: list[QGCenter],
        how: str,
        observations: list[QGPoint] | None = None,
    ) -> float:
        """Pure-Python energy fallback (no precomputed distances needed)."""
        if how == "uniform":
            energy = 0.0
            for node in self.nodes:
                neighbor = next(self.neighbors(node))
                point = QGPoint(self, (node, neighbor), 0)
                min_dist_sq = min(
                    self.distance(center, point) ** 2 for center in centers
                )
                energy += min_dist_sq
            return energy / self.number_of_nodes()
        elif how == "empirical":
            energy = sum(
                min(self.distance(center, point) ** 2 for center in centers)
                for point in observations
            )
            return energy / len(observations)
        else:  # how == "node_measure"
            energy = 0.0
            total_weight = 0
            for node, data in self.nodes(data=True):
                obs_weight = data.get("obs_weight", 0)
                if obs_weight > 0:
                    neighbor = next(self.neighbors(node))
                    point = QGPoint(self, (node, neighbor), 0)
                    min_dist_sq = min(
                        self.distance(center, point) ** 2 for center in centers
                    )
                    energy += min_dist_sq * obs_weight
                    total_weight += obs_weight

            return energy / total_weight if total_weight > 0 else 0.0

    def _points_as_arrays(
        self, points: list[QGPoint]
    ) -> tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
        """Edge indices, positions and edge lengths of points, for the kernels."""
        n = len(points)
        edges_0 = np.empty(n, dtype=np.int32)
        edges_1 = np.empty(n, dtype=np.int32)
        positions = np.empty(n, dtype=np.float64)
        lengths = np.empty(n, dtype=np.float64)
        for i, point in enumerate(points):
            edge = point.edge
            edges_0[i] = self._node_to_index[edge[0]]
            edges_1[i] = self._node_to_index[edge[1]]
            positions[i] = point.position
            lengths[i] = self.get_edge_length(*edge)
        return edges_0, edges_1, positions, lengths

    def _nodes_as_arrays(
        self,
    ) -> tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]:
        """Every node as an on-edge point at position 0, for the kernels."""
        nodes = list(self.nodes())
        n = len(nodes)
        edges_0 = np.empty(n, dtype=np.int32)
        edges_1 = np.empty(n, dtype=np.int32)
        positions = np.zeros(n, dtype=np.float64)
        lengths = np.empty(n, dtype=np.float64)
        for i, node in enumerate(nodes):
            neighbor = next(self.neighbors(node))
            edges_0[i] = self._node_to_index[node]
            edges_1[i] = self._node_to_index[neighbor]
            lengths[i] = self.get_edge_length(node, neighbor)
        return edges_0, edges_1, positions, lengths

    def calculate_energy_numba(
        self,
        centers: list[QGCenter],
        how: str = "uniform",
        observations: list[QGPoint] | None = None,
    ) -> float:
        """Numba-accelerated energy calculation for centers.

        Uses the precomputed distance matrix; ``calculate_energy`` dispatches
        here automatically once ``precomputing()`` has run.

        Args:
            centers: List of cluster centers.
            how: Reference measure — "uniform", "empirical" or "node_measure"
                (see ``calculate_energy`` for the contract; this method
                assumes the arguments were already validated there).
            observations: The points of the empirical measure
                (``how="empirical"`` only).

        Returns:
            Average squared distance to nearest center.

        Raises:
            ValueError: If pairwise distances not precomputed.
        """
        if self._pairwise_nodes_distance_array is None:
            raise ValueError("Must call precomputing() before calculate_energy_numba")

        center_arrays = self._points_as_arrays(centers)

        if how == "empirical":
            # The empirical measure of the observations: a uniform average
            # over the points themselves, exactly where they lie on their
            # edges (no rounding to nodes).
            point_arrays = self._points_as_arrays(observations)
            return _calculate_energy_uniform_numba(
                *center_arrays,
                *point_arrays,
                self._pairwise_nodes_distance_array,
            )

        point_arrays = self._nodes_as_arrays()
        if how == "uniform":
            return _calculate_energy_uniform_numba(
                *center_arrays,
                *point_arrays,
                self._pairwise_nodes_distance_array,
            )
        else:  # how == "node_measure"
            # obs_weight is a measure, not a count: integer counts come from
            # register_observations, but the paper experiments set the
            # fractional population measure nu. Casting to int here truncated
            # every fractional weight to zero, so the whole energy collapsed
            # to 0.0 -- silently, since 0 is a valid energy. Keep it float.
            point_obs_weight = np.array(
                [data.get("obs_weight", 0) for _, data in self.nodes(data=True)],
                dtype=np.float64,
            )
            return _calculate_energy_obs_numba(
                *center_arrays,
                *point_arrays,
                point_obs_weight,
                self._pairwise_nodes_distance_array,
            )

    def distance_matrix(self) -> np.ndarray:
        """Compute the pairwise distance matrix between all nodes.

        Returns:
            n×n matrix of pairwise distances.

        Raises:
            ValueError: If pairwise distances not precomputed.
        """
        if self._pairwise_nodes_distance is None:
            raise ValueError("Must call precomputing() first")

        n = self.number_of_nodes()
        node_list = list(self.nodes())
        matrix_distances = np.zeros((n, n))

        for i, node_i in enumerate(node_list):
            for j, node_j in enumerate(node_list):
                matrix_distances[i, j] = self._pairwise_nodes_distance[node_i][node_j]

        self.matrix_distance = matrix_distances
        return matrix_distances

    def index_to_centers(self, indices: list[int]) -> list[QGCenter]:
        """Convert node indices to centers.

        Args:
            indices: List of node indices.

        Returns:
            List of centers at the specified nodes.
        """
        node_list = list(self.nodes())
        return [self.node_as_center(node_list[idx]) for idx in indices]

    def draw(
        self,
        color_by: str | None = "cluster",
        centers: list[QGCenter] | None = None,
        node_size_by_obs: bool = True,
        ax: "plt.Axes" | None = None,
        **kwargs,
    ):
        """Draws the graph using matplotlib.

        This method provides a flexible way to visualize the graph, its clusters,
        and the results of the k-means algorithm. To use this method, you need
        to install the 'plot' extras: `pip install kmeanssa-ng[plot]`

        Args:
            color_by: Node attribute to use for coloring.
                - "cluster": Colors nodes by their assigned cluster.
                - "block": Colors nodes by a pre-defined 'block' attribute.
                - None: All nodes will have the same default color. (Default: "skyblue")
            centers: A list of QGCenter objects to be highlighted on the graph.
            node_size_by_obs: If True, sizes nodes based on 'obs_weight' attribute.
            ax: A matplotlib axes object to draw on. If None, a new figure
                and axes are created.
            **kwargs: Additional arguments passed to nx.draw().
        """
        try:
            import matplotlib.pyplot as plt
        except ImportError as e:
            raise ImportError(
                "Plotting requires matplotlib. Please install it with "
                "`pip install kmeanssa-ng[plot]`"
            ) from e

        if ax is None:
            fig, ax = plt.subplots(figsize=kwargs.pop("figsize", (12, 12)))

        # Use pre-computed layout for consistency
        pos = self.node_position

        # Determine node colors
        node_colors = "skyblue"  # Default color
        if color_by:
            # Check if any node has the attribute
            if not any(color_by in n[1] for n in self.nodes(data=True)):
                print(
                    f"Warning: Node attribute '{color_by}' not found. Using default color."
                )
            else:
                node_colors = [self.nodes[n].get(color_by, 0) for n in self.nodes()]

        # Determine node sizes
        node_sizes = 300  # Default size
        if node_size_by_obs:
            if not any("obs_weight" in n[1] for n in self.nodes(data=True)):
                print(
                    "Warning: Node attribute 'obs_weight' not found. Using default size."
                )
            else:
                # Scale size by number of observations, with a minimum size
                node_sizes = [
                    self.nodes[n].get("obs_weight", 0) * 50 + 100 for n in self.nodes()
                ]

        # Draw the graph
        nx.draw(
            self,
            pos=pos,
            ax=ax,
            node_color=node_colors,
            node_size=node_sizes,
            with_labels=False,
            **kwargs,
        )

        # Highlight centers
        if centers:
            center_nodes = [c.edge[0] for c in centers]
            node_list = list(self.nodes)
            center_node_indices = [node_list.index(n) for n in center_nodes]

            center_sizes = (
                [node_sizes[i] * 2 for i in center_node_indices]
                if isinstance(node_sizes, list)
                else node_sizes * 2
            )

            nx.draw_networkx_nodes(
                self,
                pos,
                nodelist=center_nodes,
                ax=ax,
                node_size=center_sizes,
                edgecolors="grey",
                linewidths=2,
            )

    def get_point_type(self) -> type[QGPoint]:
        """Return the type of points in this space."""
        return QGPoint

diameter property

Compute and cache the graph diameter.

Returns:

Type Description
float

Maximum distance between any two nodes.

node_position property

Compute layout positions for visualization.

Returns:

Type Description
dict

Dictionary mapping nodes to (x, y) positions.

__init__(incoming_graph_data=None, precompute=False, **attr)

Initialize a quantum graph.

Parameters:

Name Type Description Default
incoming_graph_data

Input graph data (see networkx.Graph).

 None
precompute bool

If True, automatically precompute distances after initialization.

 False
**attr

Additional graph attributes.

 {}
Source code in kmeanssa_ng/quantum_graph/space.py 
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def __init__(
    self, incoming_graph_data=None, precompute: bool = False, **attr
) -> None:
    """Initialize a quantum graph.

    Args:
        incoming_graph_data: Input graph data (see networkx.Graph).
        precompute: If True, automatically precompute distances after initialization.
        **attr: Additional graph attributes.
    """
    super().__init__(incoming_graph_data, **attr)
    self._pairwise_nodes_distance: dict[int, dict[int, float]] | None = None
    self._pairwise_nodes_distance_array: np.ndarray | None = None
    self._node_to_index: dict[int, int] | None = None
    self._diameter: float = 0.0
    self._node_position: dict | None = None

    # Validate any length carried by the incoming data, so a degenerate
    # edge (e.g. a zero-length edge from a collapsed epsilon-net) fails
    # here with a clear ValueError rather than silently poisoning a later
    # precomputing() when precompute=False. Only lengths that are present
    # are checked, so building the topology first and setting the lengths
    # afterwards (generate_sbm, as_quantum_graph) still works.
    for u, v, length in self.edges(data="length"):
        if length is not None:
            self._validate_length(u, v, length)

    if precompute and self.number_of_nodes() > 0:
        self.precomputing()

add_edge(u_for_edge, v_for_edge, **attr)

Add an edge with validation of the length attribute.

Invalidates the precomputed distance cache: call precomputing() again after editing the graph.

Parameters:

Name Type Description Default
u_for_edge

First node.

 required
v_for_edge

Second node.

 required
**attr

Edge attributes. Must include 'length' with a positive value.

 {}

Raises:

Type Description
ValueError

If 'length' is missing, not positive, or not a number.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def add_edge(self, u_for_edge, v_for_edge, **attr) -> None:
    """Add an edge with validation of the length attribute.

    Invalidates the precomputed distance cache: call ``precomputing()``
    again after editing the graph.

    Args:
        u_for_edge: First node.
        v_for_edge: Second node.
        **attr: Edge attributes. Must include 'length' with a positive value.

    Raises:
        ValueError: If 'length' is missing, not positive, or not a number.
    """
    if "length" not in attr:
        raise ValueError(
            f"Edge ({u_for_edge}, {v_for_edge}) must have a 'length' attribute"
        )

    self._validate_length(u_for_edge, v_for_edge, attr["length"])

    super().add_edge(u_for_edge, v_for_edge, **attr)
    self._invalidate_distance_cache()

add_edges_from(ebunch_to_add, **attr)

Add edges in bulk; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def add_edges_from(self, ebunch_to_add, **attr) -> None:
    """Add edges in bulk; invalidates the precomputed distance cache."""
    super().add_edges_from(ebunch_to_add, **attr)
    self._invalidate_distance_cache()

add_node(node_for_adding, **attr)

Add a node; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def add_node(self, node_for_adding, **attr) -> None:
    """Add a node; invalidates the precomputed distance cache."""
    super().add_node(node_for_adding, **attr)
    self._invalidate_distance_cache()

add_nodes_from(nodes_for_adding, **attr)

Add nodes in bulk; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def add_nodes_from(self, nodes_for_adding, **attr) -> None:
    """Add nodes in bulk; invalidates the precomputed distance cache."""
    super().add_nodes_from(nodes_for_adding, **attr)
    self._invalidate_distance_cache()

calculate_energy(centers, how='uniform', observations=None)

Calculate k-means energy for given centers.

Dispatches to the Numba kernels when pairwise node distances are precomputed (precomputing()), and falls back to pure-Python distance computations otherwise — correct on any graph, just slower.

Parameters:

Name Type Description Default
centers list[QGCenter]

List of cluster centers.

 required
how str

Which reference measure to average under (see :meth:Space.calculate_energy for the full contract): - "uniform": average over nodes with equal weight; rejects observations. - "empirical": average over the observations list (required), each point exactly where it lies on its edge — no rounding to nodes. - "node_measure": average over nodes weighted by their obs_weight, set explicitly by the caller (e.g. through register_observations); rejects observations.

 'uniform'
observations list[QGPoint] | None

The points defining the empirical measure (how="empirical" only).

 None

Returns:

Type Description
float

Average squared distance to nearest center.

Raises:

Type Description
ValueError

If how is unknown (including the retired "obs"), if "empirical" lacks observations, if "uniform"/"node_measure" receive some, or if no node carries a positive obs_weight in "node_measure" mode (the energy would silently be 0.0 for every configuration of centers).
Source code in kmeanssa_ng/quantum_graph/space.py 
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def calculate_energy(
    self,
    centers: list[QGCenter],
    how: str = "uniform",
    observations: list[QGPoint] | None = None,
) -> float:
    """Calculate k-means energy for given centers.

    Dispatches to the Numba kernels when pairwise node distances are
    precomputed (``precomputing()``), and falls back to pure-Python
    distance computations otherwise — correct on any graph, just slower.

    Args:
        centers: List of cluster centers.
        how: Which reference measure to average under (see
            :meth:`Space.calculate_energy` for the full contract):
            - "uniform": average over nodes with equal weight; rejects
              ``observations``.
            - "empirical": average over the ``observations`` list
              (required), each point exactly where it lies on its edge —
              no rounding to nodes.
            - "node_measure": average over nodes weighted by their
              ``obs_weight``, set explicitly by the caller (e.g. through
              ``register_observations``); rejects ``observations``.
        observations: The points defining the empirical measure
            (``how="empirical"`` only).

    Returns:
        Average squared distance to nearest center.

    Raises:
        ValueError: If ``how`` is unknown (including the retired
            ``"obs"``), if ``"empirical"`` lacks observations, if
            ``"uniform"``/``"node_measure"`` receive some, or if no node
            carries a positive ``obs_weight`` in ``"node_measure"`` mode
            (the energy would silently be 0.0 for every configuration
            of centers).
    """
    if how == "obs":
        raise ValueError(
            "energy mode 'obs' was split into two explicit modes: use "
            "'empirical' to average over an explicit list of observation "
            "points, or 'node_measure' to average under the per-node "
            "'obs_weight' measure registered on the graph."
        )
    if how not in ("uniform", "empirical", "node_measure"):
        raise ValueError(
            f"how must be 'uniform', 'empirical' or 'node_measure', got {how!r}"
        )
    if how == "empirical":
        if not observations:
            raise ValueError(
                "energy mode 'empirical' requires the explicit "
                "'observations' list defining the empirical measure"
            )
    elif observations is not None:
        raise ValueError(
            f"energy mode {how!r} defines its own reference measure and "
            "does not take 'observations'; pass them with how='empirical' "
            "instead"
        )
    if how == "node_measure" and not any(
        data.get("obs_weight", 0) > 0 for _, data in self.nodes(data=True)
    ):
        raise ValueError(
            "energy mode 'node_measure' requires a measure on the graph "
            "nodes, but no node carries a positive 'obs_weight'. Register "
            "one explicitly: register_observations(points) counts each "
            "point at its closest node, or set the 'obs_weight' node "
            "attribute directly (e.g. to a population measure)."
        )
    if self._pairwise_nodes_distance_array is not None:
        return self.calculate_energy_numba(
            centers, how=how, observations=observations
        )
    return self._calculate_energy_python(centers, how, observations=observations)

calculate_energy_numba(centers, how='uniform', observations=None)

Numba-accelerated energy calculation for centers.

Uses the precomputed distance matrix; calculate_energy dispatches here automatically once precomputing() has run.

Parameters:

Name Type Description Default
centers list[QGCenter]

List of cluster centers.

 required
how str

Reference measure — "uniform", "empirical" or "node_measure" (see calculate_energy for the contract; this method assumes the arguments were already validated there).

 'uniform'
observations list[QGPoint] | None

The points of the empirical measure (how="empirical" only).

 None

Returns:

Type Description
float

Average squared distance to nearest center.

Raises:

Type Description
ValueError

If pairwise distances not precomputed.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def calculate_energy_numba(
    self,
    centers: list[QGCenter],
    how: str = "uniform",
    observations: list[QGPoint] | None = None,
) -> float:
    """Numba-accelerated energy calculation for centers.

    Uses the precomputed distance matrix; ``calculate_energy`` dispatches
    here automatically once ``precomputing()`` has run.

    Args:
        centers: List of cluster centers.
        how: Reference measure — "uniform", "empirical" or "node_measure"
            (see ``calculate_energy`` for the contract; this method
            assumes the arguments were already validated there).
        observations: The points of the empirical measure
            (``how="empirical"`` only).

    Returns:
        Average squared distance to nearest center.

    Raises:
        ValueError: If pairwise distances not precomputed.
    """
    if self._pairwise_nodes_distance_array is None:
        raise ValueError("Must call precomputing() before calculate_energy_numba")

    center_arrays = self._points_as_arrays(centers)

    if how == "empirical":
        # The empirical measure of the observations: a uniform average
        # over the points themselves, exactly where they lie on their
        # edges (no rounding to nodes).
        point_arrays = self._points_as_arrays(observations)
        return _calculate_energy_uniform_numba(
            *center_arrays,
            *point_arrays,
            self._pairwise_nodes_distance_array,
        )

    point_arrays = self._nodes_as_arrays()
    if how == "uniform":
        return _calculate_energy_uniform_numba(
            *center_arrays,
            *point_arrays,
            self._pairwise_nodes_distance_array,
        )
    else:  # how == "node_measure"
        # obs_weight is a measure, not a count: integer counts come from
        # register_observations, but the paper experiments set the
        # fractional population measure nu. Casting to int here truncated
        # every fractional weight to zero, so the whole energy collapsed
        # to 0.0 -- silently, since 0 is a valid energy. Keep it float.
        point_obs_weight = np.array(
            [data.get("obs_weight", 0) for _, data in self.nodes(data=True)],
            dtype=np.float64,
        )
        return _calculate_energy_obs_numba(
            *center_arrays,
            *point_arrays,
            point_obs_weight,
            self._pairwise_nodes_distance_array,
        )

center_from_point(point, rng=None)

Create a QGCenter object from a QGPoint object.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def center_from_point(self, point: QGPoint, rng=None) -> QGCenter:
    """Create a QGCenter object from a QGPoint object."""
    return QGCenter(point, rng=rng)

clear()

Clear the graph; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def clear(self) -> None:
    """Clear the graph; invalidates the precomputed distance cache."""
    super().clear()
    self._invalidate_distance_cache()

clear_edges()

Clear all edges; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def clear_edges(self) -> None:
    """Clear all edges; invalidates the precomputed distance cache."""
    super().clear_edges()
    self._invalidate_distance_cache()

distance(p1, p2)

Compute distance between two points on the graph.

Parameters:

Name Type Description Default
p1 QGPoint

First point.

 required
p2 QGPoint

Second point.

 required

Returns:

Type Description
float

Geodesic distance.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def distance(self, p1: QGPoint, p2: QGPoint) -> float:
    """Compute distance between two points on the graph.

    Args:
        p1: First point.
        p2: Second point.

    Returns:
        Geodesic distance.
    """
    return self.quantum_path(p1, p2)["distance"]

distance_matrix()

Compute the pairwise distance matrix between all nodes.

Returns:

Type Description
ndarray

n×n matrix of pairwise distances.

Raises:

Type Description
ValueError

If pairwise distances not precomputed.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def distance_matrix(self) -> np.ndarray:
    """Compute the pairwise distance matrix between all nodes.

    Returns:
        n×n matrix of pairwise distances.

    Raises:
        ValueError: If pairwise distances not precomputed.
    """
    if self._pairwise_nodes_distance is None:
        raise ValueError("Must call precomputing() first")

    n = self.number_of_nodes()
    node_list = list(self.nodes())
    matrix_distances = np.zeros((n, n))

    for i, node_i in enumerate(node_list):
        for j, node_j in enumerate(node_list):
            matrix_distances[i, j] = self._pairwise_nodes_distance[node_i][node_j]

    self.matrix_distance = matrix_distances
    return matrix_distances

distances_from_centers(centers, target)

Compute distances from multiple centers to a single target point.

This is a Numba-accelerated operation that efficiently computes distances from all centers to one target point. Works for any target location (on nodes or edges).

Parameters:

Name Type Description Default
centers list[QGCenter]

List of k centers to compute distances from.

 required
target QGPoint

The target point (can be on a node or edge).

 required

Returns:

Type Description
ndarray

Array of shape (k,) with distances from each center to target.

Raises:

Type Description
ValueError

If pairwise distances not precomputed.
Example 
centers = graph.sample_centers(5)
target = graph.sample_points(1)[0]
distances = graph.distances_from_centers(centers, target)
closest_idx = np.argmin(distances)
closest_center = centers[closest_idx]
Source code in kmeanssa_ng/quantum_graph/space.py 
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def distances_from_centers(
    self, centers: list[QGCenter], target: QGPoint
) -> np.ndarray:
    """Compute distances from multiple centers to a single target point.

    This is a Numba-accelerated operation that efficiently computes distances
    from all centers to one target point. Works for any target location
    (on nodes or edges).

    Args:
        centers: List of k centers to compute distances from.
        target: The target point (can be on a node or edge).

    Returns:
        Array of shape (k,) with distances from each center to target.

    Raises:
        ValueError: If pairwise distances not precomputed.

    Example:
        ```python
        centers = graph.sample_centers(5)
        target = graph.sample_points(1)[0]
        distances = graph.distances_from_centers(centers, target)
        closest_idx = np.argmin(distances)
        closest_center = centers[closest_idx]
        ```
    """
    if self._pairwise_nodes_distance_array is None:
        raise ValueError("Must call precomputing() before distances_from_centers")

    k = len(centers)

    # Extract center data into numpy arrays for Numba
    center_edges_0 = np.empty(k, dtype=np.int32)
    center_edges_1 = np.empty(k, dtype=np.int32)
    center_positions = np.empty(k, dtype=np.float64)
    center_lengths = np.empty(k, dtype=np.float64)

    for i, center in enumerate(centers):
        edge = center.edge
        center_edges_0[i] = self._node_to_index[edge[0]]
        center_edges_1[i] = self._node_to_index[edge[1]]
        center_positions[i] = center.position
        center_lengths[i] = self.get_edge_length(*edge)

    # Extract target information
    target_edge = target.edge
    target_edge_0 = self._node_to_index[target_edge[0]]
    target_edge_1 = self._node_to_index[target_edge[1]]
    target_pos = target.position
    target_length = self.get_edge_length(*target_edge)

    # Call Numba-accelerated function
    return _batch_distances_numba(
        center_edges_0,
        center_edges_1,
        center_positions,
        center_lengths,
        target_edge_0,
        target_edge_1,
        target_pos,
        target_length,
        self._pairwise_nodes_distance_array,
    )

draw(color_by='cluster', centers=None, node_size_by_obs=True, ax=None, **kwargs)

Draws the graph using matplotlib.

This method provides a flexible way to visualize the graph, its clusters, and the results of the k-means algorithm. To use this method, you need to install the 'plot' extras: pip install kmeanssa-ng[plot]

Parameters:

Name Type Description Default
color_by str | None

Node attribute to use for coloring. - "cluster": Colors nodes by their assigned cluster. - "block": Colors nodes by a pre-defined 'block' attribute. - None: All nodes will have the same default color. (Default: "skyblue")

 'cluster'
centers list[QGCenter] | None

A list of QGCenter objects to be highlighted on the graph.

 None
node_size_by_obs bool

If True, sizes nodes based on 'obs_weight' attribute.

 True
ax 'plt.Axes' | None

A matplotlib axes object to draw on. If None, a new figure and axes are created.

 None
**kwargs

Additional arguments passed to nx.draw().

 {}
Source code in kmeanssa_ng/quantum_graph/space.py 
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def draw(
    self,
    color_by: str | None = "cluster",
    centers: list[QGCenter] | None = None,
    node_size_by_obs: bool = True,
    ax: "plt.Axes" | None = None,
    **kwargs,
):
    """Draws the graph using matplotlib.

    This method provides a flexible way to visualize the graph, its clusters,
    and the results of the k-means algorithm. To use this method, you need
    to install the 'plot' extras: `pip install kmeanssa-ng[plot]`

    Args:
        color_by: Node attribute to use for coloring.
            - "cluster": Colors nodes by their assigned cluster.
            - "block": Colors nodes by a pre-defined 'block' attribute.
            - None: All nodes will have the same default color. (Default: "skyblue")
        centers: A list of QGCenter objects to be highlighted on the graph.
        node_size_by_obs: If True, sizes nodes based on 'obs_weight' attribute.
        ax: A matplotlib axes object to draw on. If None, a new figure
            and axes are created.
        **kwargs: Additional arguments passed to nx.draw().
    """
    try:
        import matplotlib.pyplot as plt
    except ImportError as e:
        raise ImportError(
            "Plotting requires matplotlib. Please install it with "
            "`pip install kmeanssa-ng[plot]`"
        ) from e

    if ax is None:
        fig, ax = plt.subplots(figsize=kwargs.pop("figsize", (12, 12)))

    # Use pre-computed layout for consistency
    pos = self.node_position

    # Determine node colors
    node_colors = "skyblue"  # Default color
    if color_by:
        # Check if any node has the attribute
        if not any(color_by in n[1] for n in self.nodes(data=True)):
            print(
                f"Warning: Node attribute '{color_by}' not found. Using default color."
            )
        else:
            node_colors = [self.nodes[n].get(color_by, 0) for n in self.nodes()]

    # Determine node sizes
    node_sizes = 300  # Default size
    if node_size_by_obs:
        if not any("obs_weight" in n[1] for n in self.nodes(data=True)):
            print(
                "Warning: Node attribute 'obs_weight' not found. Using default size."
            )
        else:
            # Scale size by number of observations, with a minimum size
            node_sizes = [
                self.nodes[n].get("obs_weight", 0) * 50 + 100 for n in self.nodes()
            ]

    # Draw the graph
    nx.draw(
        self,
        pos=pos,
        ax=ax,
        node_color=node_colors,
        node_size=node_sizes,
        with_labels=False,
        **kwargs,
    )

    # Highlight centers
    if centers:
        center_nodes = [c.edge[0] for c in centers]
        node_list = list(self.nodes)
        center_node_indices = [node_list.index(n) for n in center_nodes]

        center_sizes = (
            [node_sizes[i] * 2 for i in center_node_indices]
            if isinstance(node_sizes, list)
            else node_sizes * 2
        )

        nx.draw_networkx_nodes(
            self,
            pos,
            nodelist=center_nodes,
            ax=ax,
            node_size=center_sizes,
            edgecolors="grey",
            linewidths=2,
        )

get_edge_length(n1, n2)

Get the length of an edge.

Parameters:

Name Type Description Default
n1 int

First node of the edge.

 required
n2 int

Second node of the edge.

 required

Returns:

Type Description
float

Edge length.

Raises:

Type Description
ValueError

If edge does not exist.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def get_edge_length(self, n1: int, n2: int) -> float:
    """Get the length of an edge.

    Args:
        n1: First node of the edge.
        n2: Second node of the edge.

    Returns:
        Edge length.

    Raises:
        ValueError: If edge does not exist.
    """
    edge_data = self.get_edge_data(n1, n2)
    if edge_data is None:
        raise ValueError(f"Edge ({n1}, {n2}) does not exist in graph")
    return edge_data["length"]

get_point_type()

Return the type of points in this space.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def get_point_type(self) -> type[QGPoint]:
    """Return the type of points in this space."""
    return QGPoint

index_to_centers(indices)

Convert node indices to centers.

Parameters:

Name Type Description Default
indices list[int]

List of node indices.

 required

Returns:

Type Description
list[QGCenter]

List of centers at the specified nodes.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def index_to_centers(self, indices: list[int]) -> list[QGCenter]:
    """Convert node indices to centers.

    Args:
        indices: List of node indices.

    Returns:
        List of centers at the specified nodes.
    """
    node_list = list(self.nodes())
    return [self.node_as_center(node_list[idx]) for idx in indices]

node_as_center(node, rng=None)

Create a center at a specific node.

Parameters:

Name Type Description Default
node int

The node to place the center at.

 required
rng

Optional random number generator for neighbor selection.

 None

Returns:

Type Description
QGCenter

A center located at the node.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def node_as_center(self, node: int, rng=None) -> QGCenter:
    """Create a center at a specific node.

    Args:
        node: The node to place the center at.
        rng: Optional random number generator for neighbor selection.

    Returns:
        A center located at the node.
    """
    if rng is not None:
        nbrs = list(self.neighbors(node))
        neighbor = nbrs[rng.integers(len(nbrs))]
    else:
        neighbor = next(self.neighbors(node))
    point = QGPoint(self, (node, neighbor), 0)
    return QGCenter(point, rng=rng)

node_center_distances(centers)

Geodesic distance from every node to every center.

Parameters:

Name Type Description Default
centers list[QGCenter]

List of k centers.

 required

Returns:

Name Type Description
ndarray

Array of shape (n_nodes, k), with rows in list(self.nodes())
order ndarray

entry (v, i) is the geodesic distance from node v to center i.

Raises:

Type Description
ValueError

If pairwise distances have not been precomputed.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def node_center_distances(self, centers: list[QGCenter]) -> np.ndarray:
    """Geodesic distance from every node to every center.

    Args:
        centers: List of k centers.

    Returns:
        Array of shape (n_nodes, k), with rows in ``list(self.nodes())``
        order: entry (v, i) is the geodesic distance from node v to center i.

    Raises:
        ValueError: If pairwise distances have not been precomputed.
    """
    if self._pairwise_nodes_distance_array is None:
        raise ValueError("Must call precomputing() before node_center_distances")

    distances = self._pairwise_nodes_distance_array  # (n, n), node order
    index = self._node_to_index
    out = np.empty((distances.shape[0], len(centers)))
    for i, center in enumerate(centers):
        a, b = center.edge
        length = self.get_edge_length(a, b)
        # A center sits at `position` along edge (a, b); a node reaches it
        # via whichever endpoint is closer.
        out[:, i] = np.minimum(
            center.position + distances[:, index[a]],
            (length - center.position) + distances[:, index[b]],
        )
    return out

node_distance(n1, n2)

Compute shortest path distance between two nodes.

Parameters:

Name Type Description Default
n1 int

First node.

 required
n2 int

Second node.

 required

Returns:

Type Description
float

Shortest path length using 'length' edge attribute.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def node_distance(self, n1: int, n2: int) -> float:
    """Compute shortest path distance between two nodes.

    Args:
        n1: First node.
        n2: Second node.

    Returns:
        Shortest path length using 'length' edge attribute.
    """
    # Prefer the precomputed array (O(1)); it may be present without the dict
    # form (e.g. loaded from a cache). Fall back to the dict, then to a live
    # Dijkstra when nothing is precomputed.
    if self._pairwise_nodes_distance_array is not None and self._node_to_index:
        return float(
            self._pairwise_nodes_distance_array[
                self._node_to_index[n1], self._node_to_index[n2]
            ]
        )
    if self._pairwise_nodes_distance is not None:
        return self._pairwise_nodes_distance[n1][n2]
    return nx.shortest_path_length(self, n1, n2, weight="length")

node_energy(centers, weights=None)

Node-weighted k-means energy sum_v w[v] * min_i d(v, c_i)^2.

Parameters:

Name Type Description Default
centers list[QGCenter]

List of cluster centers.

 required
weights ndarray | None

Per-node weights in list(self.nodes()) order. Defaults to the nodes' weight attribute (1.0 when absent).

 None

Returns:

Type Description
float

The weighted energy.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def node_energy(
    self, centers: list[QGCenter], weights: np.ndarray | None = None
) -> float:
    """Node-weighted k-means energy ``sum_v w[v] * min_i d(v, c_i)^2``.

    Args:
        centers: List of cluster centers.
        weights: Per-node weights in ``list(self.nodes())`` order. Defaults
            to the nodes' ``weight`` attribute (1.0 when absent).

    Returns:
        The weighted energy.
    """
    min_sq = self.node_center_distances(centers).min(axis=1) ** 2
    if weights is None:
        weights = np.array(
            [self.nodes[node].get("weight", 1.0) for node in self.nodes()]
        )
    return float(np.sum(np.asarray(weights) * min_sq))

nodes_as_points()

Convert all nodes to points.

Returns:

Type Description
list[QGPoint]

List of points, one at each node.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def nodes_as_points(self) -> list[QGPoint]:
    """Convert all nodes to points.

    Returns:
        List of points, one at each node.
    """
    points = []
    for node in self.nodes:
        neighbor = next(self.neighbors(node))
        points.append(QGPoint(self, (node, neighbor), 0))
    return points

precomputing()

Precompute and cache all pairwise node distances.

This significantly speeds up distance queries. Should be called once after graph construction.

Raises:

Type Description
ValueError

If graph is not connected or has invalid edge lengths.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def precomputing(self) -> None:
    """Precompute and cache all pairwise node distances.

    This significantly speeds up distance queries.
    Should be called once after graph construction.

    Raises:
        ValueError: If graph is not connected or has invalid edge lengths.
    """
    # Validate edge lengths first
    self.validate_edge_lengths()

    # Check connectivity
    if self.number_of_nodes() > 0 and not nx.is_connected(self):
        num_components = nx.number_connected_components(self)
        raise ValueError(
            f"Graph must be connected for distance precomputing. "
            f"Found {num_components} connected components."
        )

    if self._pairwise_nodes_distance is None:
        self._pairwise_nodes_distance = dict(
            nx.all_pairs_dijkstra_path_length(self, weight="length")
        )

        # Create numpy array version for Numba-accelerated computations
        node_list = list(self.nodes())
        n = len(node_list)
        self._node_to_index = {node: i for i, node in enumerate(node_list)}
        self._pairwise_nodes_distance_array = np.zeros((n, n), dtype=np.float64)

        for i, node_i in enumerate(node_list):
            for j, node_j in enumerate(node_list):
                self._pairwise_nodes_distance_array[i, j] = (
                    self._pairwise_nodes_distance[node_i][node_j]
                )

quantum_path(p1, p2)

Compute the geodesic between two points on the graph.

Parameters:

Name Type Description Default
p1 QGPoint

First point.

 required
p2 QGPoint

Second point.

 required

Returns:

Type Description
dict[str, float | tuple | None]

Dictionary with: - 'distance': geodesic distance - 'path': (node_from_p1_edge, node_from_p2_edge) or None if same edge
Source code in kmeanssa_ng/quantum_graph/space.py 
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def quantum_path(self, p1: QGPoint, p2: QGPoint) -> dict[str, float | tuple | None]:
    """Compute the geodesic between two points on the graph.

    Args:
        p1: First point.
        p2: Second point.

    Returns:
        Dictionary with:
            - 'distance': geodesic distance
            - 'path': (node_from_p1_edge, node_from_p2_edge) or None if same edge
    """
    edge1, edge2 = p1.edge, p2.edge
    pos1, pos2 = p1.position, p2.position
    length1 = self.get_edge_length(*edge1)
    length2 = self.get_edge_length(*edge2)

    # Both points on the same self-loop: the geodesic stays on the loop
    # (direct arc or the complementary one, possibly through the vertex).
    if edge1[0] == edge1[1] and edge2[0] == edge2[1] and edge1[0] == edge2[0]:
        arc = abs(pos1 - pos2)
        return {"distance": min(arc, length1 - arc), "path": None}

    # Compute distances for all 4 possible paths
    d0 = self.node_distance(edge1[0], edge2[0]) + pos1 + pos2
    d1 = self.node_distance(edge1[0], edge2[1]) + pos1 + (length2 - pos2)
    d2 = self.node_distance(edge1[1], edge2[0]) + (length1 - pos1) + pos2
    d3 = (
        self.node_distance(edge1[1], edge2[1]) + (length1 - pos1) + (length2 - pos2)
    )

    # Find minimum distance (break ties deterministically)
    distances = np.array([d0, d1, d2, d3])
    all_idx = np.where(distances == distances.min())[0]
    idx = all_idx[0]
    d_min = distances[idx]

    # Check if points are on the same (non-loop) physical edge
    if (
        edge1[0] != edge1[1]
        and edge1[0] == edge2[1]
        and edge1[1] == edge2[0]
        and d_min > abs(length1 - pos1 - pos2)
    ):
        return {"distance": abs(length1 - pos1 - pos2), "path": None}

    if edge1[0] != edge1[1] and edge1 == edge2 and abs(pos1 - pos2) < d_min:
        return {"distance": abs(pos1 - pos2), "path": None}

    # Determine path nodes
    if idx == 0:
        path = (edge1[0], edge2[0])
    elif idx == 1:
        path = (edge1[0], edge2[1])
    elif idx == 2:
        path = (edge1[1], edge2[0])
    else:
        path = (edge1[1], edge2[1])

    return {"distance": d_min, "path": path}

register_observations(points)

Set the per-node observation measure obs_weight from these points.

Each point counts at its closest node. This replaces any previous measure — the measure describes one observation set — so callers combining several samples must register the union explicitly.

Parameters:

Name Type Description Default
points list[QGPoint]

The observation points (any points of this graph).

 required
Source code in kmeanssa_ng/quantum_graph/space.py 
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def register_observations(self, points: list[QGPoint]) -> None:
    """Set the per-node observation measure ``obs_weight`` from these points.

    Each point counts at its closest node. This **replaces** any previous
    measure — the measure describes one observation set — so callers
    combining several samples must register the union explicitly.

    Args:
        points: The observation points (any points of this graph).
    """
    nx.set_node_attributes(self, 0, "obs_weight")
    counts: dict = {}
    for point in points:
        node = point.closest_node()
        counts[node] = counts.get(node, 0) + 1
    nx.set_node_attributes(self, counts, "obs_weight")

remove_edge(u, v)

Remove an edge; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def remove_edge(self, u, v) -> None:
    """Remove an edge; invalidates the precomputed distance cache."""
    super().remove_edge(u, v)
    self._invalidate_distance_cache()

remove_edges_from(ebunch)

Remove edges in bulk; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def remove_edges_from(self, ebunch) -> None:
    """Remove edges in bulk; invalidates the precomputed distance cache."""
    super().remove_edges_from(ebunch)
    self._invalidate_distance_cache()

remove_node(n)

Remove a node; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def remove_node(self, n) -> None:
    """Remove a node; invalidates the precomputed distance cache."""
    super().remove_node(n)
    self._invalidate_distance_cache()

remove_nodes_from(nodes)

Remove nodes in bulk; invalidates the precomputed distance cache.

Source code in kmeanssa_ng/quantum_graph/space.py 
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def remove_nodes_from(self, nodes) -> None:
    """Remove nodes in bulk; invalidates the precomputed distance cache."""
    super().remove_nodes_from(nodes)
    self._invalidate_distance_cache()

validate_edge_lengths()

Validate that all edges have positive length attributes.

Raises:

Type Description
ValueError

If any edge is missing 'length' or has invalid length.
Source code in kmeanssa_ng/quantum_graph/space.py 
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def validate_edge_lengths(self) -> None:
    """Validate that all edges have positive length attributes.

    Raises:
        ValueError: If any edge is missing 'length' or has invalid length.
    """
    for u, v, data in self.edges(data=True):
        if "length" not in data:
            raise ValueError(f"Edge ({u}, {v}) missing 'length' attribute")

        length = data["length"]
        try:
            length_float = float(length)
        except (TypeError, ValueError) as e:
            raise ValueError(
                f"Edge ({u}, {v}) length must be a number, got {type(length).__name__}"
            ) from e

        if length_float <= 0:
            raise ValueError(
                f"Edge ({u}, {v}) has invalid length {length_float}, must be positive"
            )

as_quantum_graph(graph, node_weight=1.0, edge_length=1.0, edge_weight=1.0, precompute=False)

Convert a NetworkX graph to a quantum graph with uniform attributes.

Parameters:

Name Type Description Default
graph Graph

The NetworkX graph to convert.

 required
node_weight float

Uniform weight to assign to all nodes.

 1.0
edge_length float

Uniform length to assign to all edges.

 1.0
edge_weight float

Uniform weight to assign to all edges.

 1.0
precompute bool

If True, precompute pairwise distances (default: False for compatibility).

 False

Returns:

Type Description
QuantumGraph

The converted quantum graph.
Example 
import networkx as nx
G = nx.karate_club_graph()
qg = as_quantum_graph(G, edge_length=1.0)
Source code in kmeanssa_ng/quantum_graph/generators.py 
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def as_quantum_graph(
    graph: nx.Graph,
    node_weight: float = 1.0,
    edge_length: float = 1.0,
    edge_weight: float = 1.0,
    precompute: bool = False,
) -> QuantumGraph:
    """Convert a NetworkX graph to a quantum graph with uniform attributes.

    Args:
        graph: The NetworkX graph to convert.
        node_weight: Uniform weight to assign to all nodes.
        edge_length: Uniform length to assign to all edges.
        edge_weight: Uniform weight to assign to all edges.
        precompute: If True, precompute pairwise distances (default: False for compatibility).

    Returns:
        The converted quantum graph.

    Example:
        ```python
        import networkx as nx
        G = nx.karate_club_graph()
        qg = as_quantum_graph(G, edge_length=1.0)
        ```
    """
    # Validate 'graph'
    if not isinstance(graph, nx.Graph):
        raise ValueError("`graph` must be a networkx.Graph object.")

    # Validate 'node_weight'
    if not isinstance(node_weight, (int, float)) or node_weight <= 0:
        raise ValueError("`node_weight` must be a positive number.")

    # Validate 'edge_length'
    if not isinstance(edge_length, (int, float)) or edge_length <= 0:
        raise ValueError("`edge_length` must be a positive number.")

    # Validate 'edge_weight'
    if not isinstance(edge_weight, (int, float)) or edge_weight <= 0:
        raise ValueError("`edge_weight` must be a positive number.")

    qg = QuantumGraph(graph, precompute=False)
    nx.set_node_attributes(qg, node_weight, "weight")
    nx.set_edge_attributes(qg, edge_length, "length")
    nx.set_edge_attributes(qg, edge_weight, "weight")

    if precompute:
        qg.precomputing()
    return qg

complete_quantum_graph(objects, similarities=None, true_labels=None, precompute=True)

Create a complete quantum graph from objects with optional similarity matrix.

Useful for clustering when you have a pairwise distance/similarity matrix.

Parameters:

Name Type Description Default
objects list

List of objects (nodes will be indexed by position).

 required
similarities ndarray | None

Optional n×n matrix of similarities/distances. If None, all edges have length 1.

 None
true_labels list | None

Optional true cluster labels for each object.

 None
precompute bool

If True, precompute pairwise distances (default: True).

 True

Returns:

Type Description
QuantumGraph

A complete quantum graph where edge lengths are given by the similarity matrix.
Example 
objects = [1, 2, 3, 4, 5]
similarities = np.array([...])  # 5x5 matrix
labels = [0, 0, 1, 1, 1]
graph = complete_quantum_graph(objects, similarities, labels)
Source code in kmeanssa_ng/quantum_graph/generators.py 
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def complete_quantum_graph(
    objects: list,
    similarities: np.ndarray | None = None,
    true_labels: list | None = None,
    precompute: bool = True,
) -> QuantumGraph:
    """Create a complete quantum graph from objects with optional similarity matrix.

    Useful for clustering when you have a pairwise distance/similarity matrix.

    Args:
        objects: List of objects (nodes will be indexed by position).
        similarities: Optional n×n matrix of similarities/distances. If None, all edges
            have length 1.
        true_labels: Optional true cluster labels for each object.
        precompute: If True, precompute pairwise distances (default: True).

    Returns:
        A complete quantum graph where edge lengths are given by the similarity matrix.

    Example:
        ```python
        objects = [1, 2, 3, 4, 5]
        similarities = np.array([...])  # 5x5 matrix
        labels = [0, 0, 1, 1, 1]
        graph = complete_quantum_graph(objects, similarities, labels)
        ```
    """
    # Validate 'objects'
    if not isinstance(objects, list) or not objects:
        raise ValueError("`objects` must be a non-empty list.")

    num_objects = len(objects)

    # Validate 'similarities'
    if similarities is not None:
        if not isinstance(similarities, np.ndarray):
            raise ValueError("`similarities` must be a numpy array.")
        if similarities.shape != (num_objects, num_objects):
            raise ValueError(
                f"`similarities` must be a square matrix of size {num_objects}x{num_objects}."
            )
        # Edge lengths are the off-diagonal similarities; they become edge
        # lengths, which must be strictly positive (a zero would be a
        # zero-length edge). Also require symmetry, since the graph is
        # undirected and only the upper triangle is read.
        off_diagonal = ~np.eye(num_objects, dtype=bool)
        if np.any(similarities[off_diagonal] <= 0):
            raise ValueError(
                "Off-diagonal `similarities` must be strictly positive: they "
                "become edge lengths, and a zero or negative value is not a "
                "valid length."
            )
        if not np.allclose(similarities, similarities.T):
            raise ValueError("`similarities` must be symmetric.")

    # Validate 'true_labels'
    if true_labels is not None:
        if not isinstance(true_labels, list):
            raise ValueError("`true_labels` must be a list.")
        if len(true_labels) != num_objects:
            raise ValueError(
                f"`true_labels` must have the same length as `objects` ({num_objects})."
            )

    graph = QuantumGraph(precompute=False)

    # Add nodes
    for i, _ in enumerate(objects):
        if true_labels is not None:
            graph.add_node(i, weight=1, group=true_labels[i])
        else:
            graph.add_node(i, weight=1)

    # Add all edges (complete graph)
    for i in range(len(objects)):
        for j in range(i + 1, len(objects)):
            if similarities is not None:
                edge_length = similarities[i][j]
            else:
                edge_length = 1.0

            graph.add_edge(i, j, weight=1, length=edge_length)

    if precompute:
        graph.precomputing()
    return graph

generate_random_sbm(sizes=None, p=None, weights=None, lengths=None, precompute=True, random_state=None)

Generate an SBM quantum graph with block-specific edge lengths and node weights.

Parameters:

Name Type Description Default
sizes list[int] | None

Number of nodes in each block. Defaults to [50, 50].

 None
p list[list[float]] | None

Matrix of edge probabilities. Defaults to [[0.7, 0.1], [0.1, 0.7]].

 None
weights list[float] | None

Node weight for each block. Defaults to [1, 1].

 None
lengths list[list[float]] | None

Matrix of edge lengths. Element (i, j) gives the length for edges between blocks i and j. Defaults to [[1, 4], [4, 1]].

 None
precompute bool

If True, precompute pairwise distances (default: True).

 True
random_state int | Generator | None

Seed or Generator controlling the block-model edge draws, for a reproducible graph structure. None (default) is random.

 None

Returns:

Type Description
QuantumGraph

A quantum graph with block-specific attributes.
Example 
# Two clusters with different intra/inter-cluster distances
graph = generate_random_sbm(
    sizes=[50, 50],
    p=[[0.7, 0.1], [0.1, 0.7]],
    weights=[1, 1],
    lengths=[[1, 4], [4, 1]]  # Longer inter-cluster edges
)
Source code in kmeanssa_ng/quantum_graph/generators.py 
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def generate_random_sbm(
    sizes: list[int] | None = None,
    p: list[list[float]] | None = None,
    weights: list[float] | None = None,
    lengths: list[list[float]] | None = None,
    precompute: bool = True,
    random_state: int | np.random.Generator | None = None,
) -> QuantumGraph:
    """Generate an SBM quantum graph with block-specific edge lengths and node weights.

    Args:
        sizes: Number of nodes in each block. Defaults to [50, 50].
        p: Matrix of edge probabilities. Defaults to [[0.7, 0.1], [0.1, 0.7]].
        weights: Node weight for each block. Defaults to [1, 1].
        lengths: Matrix of edge lengths. Element (i, j) gives the length
            for edges between blocks i and j. Defaults to [[1, 4], [4, 1]].
        precompute: If True, precompute pairwise distances (default: True).
        random_state: Seed or Generator controlling the block-model edge draws,
            for a reproducible graph structure. None (default) is random.

    Returns:
        A quantum graph with block-specific attributes.

    Example:
        ```python
        # Two clusters with different intra/inter-cluster distances
        graph = generate_random_sbm(
            sizes=[50, 50],
            p=[[0.7, 0.1], [0.1, 0.7]],
            weights=[1, 1],
            lengths=[[1, 4], [4, 1]]  # Longer inter-cluster edges
        )
        ```
    """
    if sizes is None:
        sizes = [50, 50]
    if p is None:
        p = [[0.7, 0.1], [0.1, 0.7]]
    if weights is None:
        weights = [1, 1]
    if lengths is None:
        lengths = [[1, 4], [4, 1]]

    num_blocks = len(sizes)

    # Validate 'sizes'
    if (
        not isinstance(sizes, list)
        or not sizes
        or not all(isinstance(s, int) and s > 0 for s in sizes)
    ):
        raise ValueError("`sizes` must be a non-empty list of positive integers.")

    # Validate 'p'
    if not isinstance(p, list) or len(p) != num_blocks:
        raise ValueError(
            f"`p` must be a square matrix of size {num_blocks}x{num_blocks}."
        )
    for row in p:
        if not isinstance(row, list) or len(row) != num_blocks:
            raise ValueError(
                f"`p` must be a square matrix of size {num_blocks}x{num_blocks}."
            )
        if not all(isinstance(val, (float, int)) and 0 <= val <= 1 for val in row):
            raise ValueError(
                "Elements of `p` must be floats or integers between 0 and 1."
            )

    # Validate 'weights'
    if not isinstance(weights, list) or len(weights) != num_blocks:
        raise ValueError(f"`weights` must be a list of size {num_blocks}.")
    if not all(isinstance(w, (float, int)) and w > 0 for w in weights):
        raise ValueError("Elements of `weights` must be positive numbers.")

    # Validate 'lengths'
    if not isinstance(lengths, list) or len(lengths) != num_blocks:
        raise ValueError(
            f"`lengths` must be a square matrix of size {num_blocks}x{num_blocks}."
        )
    for row in lengths:
        if not isinstance(row, list) or len(row) != num_blocks:
            raise ValueError(
                f"`lengths` must be a square matrix of size {num_blocks}x{num_blocks}."
            )
        if not all(isinstance(val, (float, int)) and val > 0 for val in row):
            raise ValueError("Elements of `lengths` must be positive numbers.")

    nx_graph = nx.stochastic_block_model(sizes=sizes, p=p, seed=random_state)
    graph = QuantumGraph(nx_graph, precompute=False)

    # Set node weights based on block
    for node in graph.nodes:
        block_idx = nx.get_node_attributes(graph, name="block")[node]
        w = weights[block_idx]
        nx.set_node_attributes(graph, {node: {"weight": w}})

    # Set edge lengths based on blocks
    for edge in graph.edges:
        block_i = nx.get_node_attributes(graph, name="block")[edge[0]]
        block_j = nx.get_node_attributes(graph, name="block")[edge[1]]
        edge_length = lengths[block_i][block_j]

        nx.set_edge_attributes(graph, {edge: {"length": edge_length, "weight": 1}})

    if precompute:
        graph.precomputing()
    return graph

generate_sbm(sizes=None, p=None, precompute=True, random_state=None)

Generate a Stochastic Block Model quantum graph.

Creates a quantum graph from a stochastic block model with uniform edge lengths and node weights.

Parameters:

Name Type Description Default
sizes list[int] | None

Number of nodes in each block. Defaults to [50, 50]. Must be a non-empty list of positive integers.

 None
p list[list[float]] | None

Matrix of edge probabilities. Element (r, s) gives the density of edges from block r to block s. Must be symmetric for undirected graphs. Defaults to [[0.7, 0.1], [0.1, 0.7]]. Must be a square matrix with probabilities in [0, 1].

 None
precompute bool

If True, precompute pairwise distances (default: True).

 True
random_state int | Generator | None

Seed or Generator controlling the block-model edge draws, for a reproducible graph structure. None (default) is random.

 None

Returns:

Type Description
QuantumGraph

A quantum graph representing the SBM.

Raises:

Type Description
ValueError

If sizes is empty, contains non-positive values, or if p is not a valid probability matrix matching sizes.
Example 
# Two balanced clusters with high intra-cluster, low inter-cluster edges
graph = generate_sbm(sizes=[50, 50], p=[[0.7, 0.1], [0.1, 0.7]])
Source code in kmeanssa_ng/quantum_graph/generators.py 
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def generate_sbm(
    sizes: list[int] | None = None,
    p: list[list[float]] | None = None,
    precompute: bool = True,
    random_state: int | np.random.Generator | None = None,
) -> QuantumGraph:
    """Generate a Stochastic Block Model quantum graph.

    Creates a quantum graph from a stochastic block model with uniform
    edge lengths and node weights.

    Args:
        sizes: Number of nodes in each block. Defaults to [50, 50].
            Must be a non-empty list of positive integers.
        p: Matrix of edge probabilities. Element (r, s) gives the density
            of edges from block r to block s. Must be symmetric for undirected graphs.
            Defaults to [[0.7, 0.1], [0.1, 0.7]].
            Must be a square matrix with probabilities in [0, 1].
        precompute: If True, precompute pairwise distances (default: True).
        random_state: Seed or Generator controlling the block-model edge draws,
            for a reproducible graph structure. None (default) is random.

    Returns:
        A quantum graph representing the SBM.

    Raises:
        ValueError: If sizes is empty, contains non-positive values,
            or if p is not a valid probability matrix matching sizes.

    Example:
        ```python
        # Two balanced clusters with high intra-cluster, low inter-cluster edges
        graph = generate_sbm(sizes=[50, 50], p=[[0.7, 0.1], [0.1, 0.7]])
        ```
    """
    if sizes is None:
        sizes = [50, 50]
    if p is None:
        p = [[0.7, 0.1], [0.1, 0.7]]

    # Validate sizes
    if not isinstance(sizes, list) or len(sizes) == 0:
        raise ValueError("sizes must be a non-empty list")

    for i, size in enumerate(sizes):
        try:
            size_int = int(size)
        except (TypeError, ValueError) as e:
            raise ValueError(
                f"sizes[{i}] must be an integer, got {type(size).__name__}"
            ) from e
        if size_int <= 0:
            raise ValueError(f"sizes[{i}] must be positive, got {size_int}")

    # Validate p
    if not isinstance(p, list) or len(p) == 0:
        raise ValueError("p must be a non-empty list")

    if len(p) != len(sizes):
        raise ValueError(f"p must have {len(sizes)} rows to match sizes, got {len(p)}")

    for i, row in enumerate(p):
        if not isinstance(row, list):
            raise ValueError(f"p[{i}] must be a list, got {type(row).__name__}")
        if len(row) != len(sizes):
            raise ValueError(f"p[{i}] must have {len(sizes)} columns, got {len(row)}")

        for j, prob in enumerate(row):
            try:
                prob_float = float(prob)
            except (TypeError, ValueError) as e:
                raise ValueError(
                    f"p[{i}][{j}] must be a number, got {type(prob).__name__}"
                ) from e
            if prob_float < 0 or prob_float > 1:
                raise ValueError(f"p[{i}][{j}] must be in [0, 1], got {prob_float}")

    nx_graph = nx.stochastic_block_model(sizes=sizes, p=p, seed=random_state)
    graph = QuantumGraph(
        nx_graph, precompute=False, attr=nx.get_node_attributes(nx_graph, name="block")
    )

    # Set uniform attributes
    nx.set_node_attributes(graph, 1, "weight")
    nx.set_edge_attributes(graph, 1, "length")

    for edge in graph.edges:
        nx.set_edge_attributes(graph, {edge: {"weight": 1}})

    if precompute:
        graph.precomputing()
    return graph

generate_simple_graph(n_a=5, n_aa=3, bridge_length=2.0, precompute=True, **attr)

Generate a symmetric two-cluster graph connected by a bridge.

Creates a graph with two symmetric star-like clusters (A and B) connected by a single edge. Each cluster has a central node with n_a neighbors, and each neighbor has n_aa further neighbors.

Parameters:

Name Type Description Default
n_a int

Number of neighbors for each central node (must be >= 0).

 5
n_aa int

Number of second-level neighbors (must be >= 0).

 3
bridge_length float

Length of the edge connecting the two clusters (must be > 0).

 2.0
precompute bool

If True, precompute pairwise distances (default: True).

 True
**attr

Additional graph attributes.

 {}

Returns:

Type Description
QuantumGraph

A quantum graph with two symmetric clusters.

Raises:

Type Description
ValueError

If n_a, n_aa < 0 or bridge_length <= 0.
Example 
graph = generate_simple_graph(n_a=5, n_aa=3, bridge_length=2.0)
Source code in kmeanssa_ng/quantum_graph/generators.py 
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def generate_simple_graph(
    n_a: int = 5,
    n_aa: int = 3,
    bridge_length: float = 2.0,
    precompute: bool = True,
    **attr,
) -> QuantumGraph:
    """Generate a symmetric two-cluster graph connected by a bridge.

    Creates a graph with two symmetric star-like clusters (A and B) connected
    by a single edge. Each cluster has a central node with n_a neighbors,
    and each neighbor has n_aa further neighbors.

    Args:
        n_a: Number of neighbors for each central node (must be >= 0).
        n_aa: Number of second-level neighbors (must be >= 0).
        bridge_length: Length of the edge connecting the two clusters (must be > 0).
        precompute: If True, precompute pairwise distances (default: True).
        **attr: Additional graph attributes.

    Returns:
        A quantum graph with two symmetric clusters.

    Raises:
        ValueError: If n_a, n_aa < 0 or bridge_length <= 0.

    Example:
        ```python
        graph = generate_simple_graph(n_a=5, n_aa=3, bridge_length=2.0)
        ```
    """
    # Validate n_a
    try:
        n_a_int = int(n_a)
    except (TypeError, ValueError) as e:
        raise ValueError(f"n_a must be an integer, got {type(n_a).__name__}") from e
    if n_a_int < 0:
        raise ValueError(f"n_a must be non-negative, got {n_a_int}")

    # Validate n_aa
    try:
        n_aa_int = int(n_aa)
    except (TypeError, ValueError) as e:
        raise ValueError(f"n_aa must be an integer, got {type(n_aa).__name__}") from e
    if n_aa_int < 0:
        raise ValueError(f"n_aa must be non-negative, got {n_aa_int}")

    # Validate bridge_length
    try:
        bridge_length_float = float(bridge_length)
    except (TypeError, ValueError) as e:
        raise ValueError(
            f"bridge_length must be a number, got {type(bridge_length).__name__}"
        ) from e
    if bridge_length_float <= 0:
        raise ValueError(f"bridge_length must be positive, got {bridge_length_float}")

    graph = QuantumGraph(precompute=False, **attr)

    # Add central nodes
    graph.add_node("A0", weight=1)
    graph.add_node("B0", weight=1)
    graph.add_edge("A0", "B0", length=bridge_length_float)

    # Build cluster A
    for i in range(1, n_a_int + 1):
        node_a = f"A{i}"
        graph.add_node(node_a, weight=1)
        graph.add_edge("A0", node_a, length=1.0)

        for j in range(1, n_aa_int + 1):
            node_aa = f"{node_a}_{j}"
            graph.add_node(node_aa, weight=1)
            graph.add_edge(node_a, node_aa, length=1.0)

    # Build cluster B (symmetric to A)
    for i in range(1, n_a_int + 1):
        node_b = f"B{i}"
        graph.add_node(node_b, weight=1)
        graph.add_edge("B0", node_b, length=1.0)

        for j in range(1, n_aa_int + 1):
            node_bb = f"{node_b}_{j}"
            graph.add_node(node_bb, weight=1)
            graph.add_edge(node_b, node_bb, length=1.0)

    # Set edge weights.
    for edge in graph.edges:
        nx.set_edge_attributes(graph, {edge: {"weight": 1.0}})

    if precompute:
        graph.precomputing()
    return graph

generate_simple_random_graph(n_a=5, n_b=5, lam_a=0, lam_b=0, bridge_length=10.0, precompute=True, random_state=None, **attr)

Generate a random two-cluster graph with Poisson branching.

Similar to generate_simple_graph but with: - Asymmetric clusters (different sizes) - Random edge lengths - Poisson-distributed third-level neighbors

Parameters:

Name Type Description Default
n_a int

Number of first-level neighbors of A0.

 5
n_b int

Number of first-level neighbors of B0.

 5
lam_a int

Poisson parameter for A cluster third-level branching.

 0
lam_b int

Poisson parameter for B cluster third-level branching.

 0
bridge_length float

Mean length of the bridge edge (actual length is uniform random).

 10.0
precompute bool

If True, precompute pairwise distances (default: True).

 True
random_state int | Generator | None

Seed or Generator controlling edge lengths and Poisson branching, for reproducible graphs. None (default) is random.

 None
**attr

Additional graph attributes.

 {}

Returns:

Type Description
QuantumGraph

A random quantum graph with two clusters.
Source code in kmeanssa_ng/quantum_graph/generators.py 
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def generate_simple_random_graph(
    n_a: int = 5,
    n_b: int = 5,
    lam_a: int = 0,
    lam_b: int = 0,
    bridge_length: float = 10.0,
    precompute: bool = True,
    random_state: int | np.random.Generator | None = None,
    **attr,
) -> QuantumGraph:
    """Generate a random two-cluster graph with Poisson branching.

    Similar to generate_simple_graph but with:
    - Asymmetric clusters (different sizes)
    - Random edge lengths
    - Poisson-distributed third-level neighbors

    Args:
        n_a: Number of first-level neighbors of A0.
        n_b: Number of first-level neighbors of B0.
        lam_a: Poisson parameter for A cluster third-level branching.
        lam_b: Poisson parameter for B cluster third-level branching.
        bridge_length: Mean length of the bridge edge (actual length is uniform random).
        precompute: If True, precompute pairwise distances (default: True).
        random_state: Seed or Generator controlling edge lengths and Poisson
            branching, for reproducible graphs. None (default) is random.
        **attr: Additional graph attributes.

    Returns:
        A random quantum graph with two clusters.
    """
    graph = QuantumGraph(precompute=False, **attr)
    rng = np.random.default_rng(random_state)

    # Central nodes and bridge
    graph.add_node("A0", weight=5)
    graph.add_node("B0", weight=5)
    graph.add_edge(
        "A0", "B0", length=rng.uniform(0.9 * bridge_length, 1.1 * bridge_length)
    )

    # Build cluster A
    for i in range(1, n_a + 1):
        node_a = f"A{i}"
        graph.add_node(node_a, weight=3)
        graph.add_edge("A0", node_a, length=rng.uniform(0.9, 1.1))

        # Poisson-distributed third level
        num_children = rng.poisson(lam_a)
        for j in range(1, num_children + 1):
            node_aa = f"{node_a}_{j}"
            graph.add_node(node_aa, weight=1)
            graph.add_edge(node_a, node_aa, length=rng.uniform(0.4, 0.6))

    # Build cluster B
    for i in range(1, n_b + 1):
        node_b = f"B{i}"
        graph.add_node(node_b, weight=3)
        graph.add_edge("B0", node_b, length=rng.uniform(0.9, 1.1))

        num_children = rng.poisson(lam_b)
        for j in range(1, num_children + 1):
            node_bb = f"{node_b}_{j}"
            graph.add_node(node_bb, weight=1)
            graph.add_edge(node_b, node_bb, length=rng.uniform(0.4, 0.6))

    # Set edge weights to the harmonic mean of the endpoint node weights.
    node_weights = nx.get_node_attributes(graph, "weight")
    for edge in graph.edges:
        w = 2.0 / (1.0 / node_weights[edge[0]] + 1.0 / node_weights[edge[1]])
        nx.set_edge_attributes(graph, {edge: {"weight": w}})

    if precompute:
        graph.precomputing()
    return graph

:::