229 lines
8.0 KiB
Python
229 lines
8.0 KiB
Python
"""Current-flow betweenness centrality measures for subsets of nodes."""
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import networkx as nx
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from networkx.algorithms.centrality.flow_matrix import flow_matrix_row
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from networkx.utils import not_implemented_for, reverse_cuthill_mckee_ordering
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__all__ = [
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"current_flow_betweenness_centrality_subset",
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"edge_current_flow_betweenness_centrality_subset",
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]
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@not_implemented_for("directed")
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def current_flow_betweenness_centrality_subset(
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G, sources, targets, normalized=True, weight=None, dtype=float, solver="lu"
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):
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r"""Compute current-flow betweenness centrality for subsets of nodes.
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Current-flow betweenness centrality uses an electrical current
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model for information spreading in contrast to betweenness
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centrality which uses shortest paths.
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Current-flow betweenness centrality is also known as
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random-walk betweenness centrality [2]_.
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Parameters
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----------
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G : graph
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A NetworkX graph
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sources: list of nodes
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Nodes to use as sources for current
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targets: list of nodes
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Nodes to use as sinks for current
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normalized : bool, optional (default=True)
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If True the betweenness values are normalized by b=b/(n-1)(n-2) where
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n is the number of nodes in G.
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weight : string or None, optional (default=None)
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Key for edge data used as the edge weight.
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If None, then use 1 as each edge weight.
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dtype: data type (float)
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Default data type for internal matrices.
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Set to np.float32 for lower memory consumption.
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solver: string (default='lu')
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Type of linear solver to use for computing the flow matrix.
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Options are "full" (uses most memory), "lu" (recommended), and
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"cg" (uses least memory).
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Returns
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-------
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nodes : dictionary
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Dictionary of nodes with betweenness centrality as the value.
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See Also
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--------
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approximate_current_flow_betweenness_centrality
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betweenness_centrality
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edge_betweenness_centrality
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edge_current_flow_betweenness_centrality
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Notes
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-----
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Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$
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time [1]_, where $I(n-1)$ is the time needed to compute the
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inverse Laplacian. For a full matrix this is $O(n^3)$ but using
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sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the
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Laplacian matrix condition number.
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The space required is $O(nw)$ where $w$ is the width of the sparse
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Laplacian matrix. Worse case is $w=n$ for $O(n^2)$.
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If the edges have a 'weight' attribute they will be used as
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weights in this algorithm. Unspecified weights are set to 1.
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References
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----------
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.. [1] Centrality Measures Based on Current Flow.
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Ulrik Brandes and Daniel Fleischer,
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Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
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LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
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http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf
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.. [2] A measure of betweenness centrality based on random walks,
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M. E. J. Newman, Social Networks 27, 39-54 (2005).
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"""
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from networkx.utils import reverse_cuthill_mckee_ordering
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try:
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import numpy as np
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except ImportError as e:
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raise ImportError(
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"current_flow_betweenness_centrality requires NumPy ", "http://numpy.org/"
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) from e
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if not nx.is_connected(G):
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raise nx.NetworkXError("Graph not connected.")
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n = G.number_of_nodes()
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ordering = list(reverse_cuthill_mckee_ordering(G))
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# make a copy with integer labels according to rcm ordering
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# this could be done without a copy if we really wanted to
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mapping = dict(zip(ordering, range(n)))
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H = nx.relabel_nodes(G, mapping)
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betweenness = dict.fromkeys(H, 0.0) # b[v]=0 for v in H
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for row, (s, t) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver):
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for ss in sources:
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i = mapping[ss]
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for tt in targets:
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j = mapping[tt]
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betweenness[s] += 0.5 * np.abs(row[i] - row[j])
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betweenness[t] += 0.5 * np.abs(row[i] - row[j])
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if normalized:
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nb = (n - 1.0) * (n - 2.0) # normalization factor
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else:
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nb = 2.0
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for v in H:
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betweenness[v] = betweenness[v] / nb + 1.0 / (2 - n)
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return {ordering[k]: v for k, v in betweenness.items()}
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@not_implemented_for("directed")
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def edge_current_flow_betweenness_centrality_subset(
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G, sources, targets, normalized=True, weight=None, dtype=float, solver="lu"
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):
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r"""Compute current-flow betweenness centrality for edges using subsets
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of nodes.
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Current-flow betweenness centrality uses an electrical current
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model for information spreading in contrast to betweenness
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centrality which uses shortest paths.
|
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|
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Current-flow betweenness centrality is also known as
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random-walk betweenness centrality [2]_.
|
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Parameters
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----------
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G : graph
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A NetworkX graph
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sources: list of nodes
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Nodes to use as sources for current
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targets: list of nodes
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Nodes to use as sinks for current
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normalized : bool, optional (default=True)
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If True the betweenness values are normalized by b=b/(n-1)(n-2) where
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n is the number of nodes in G.
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weight : string or None, optional (default=None)
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Key for edge data used as the edge weight.
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If None, then use 1 as each edge weight.
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dtype: data type (float)
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Default data type for internal matrices.
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Set to np.float32 for lower memory consumption.
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solver: string (default='lu')
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Type of linear solver to use for computing the flow matrix.
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Options are "full" (uses most memory), "lu" (recommended), and
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"cg" (uses least memory).
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Returns
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-------
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nodes : dict
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Dictionary of edge tuples with betweenness centrality as the value.
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See Also
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--------
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betweenness_centrality
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edge_betweenness_centrality
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current_flow_betweenness_centrality
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Notes
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-----
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Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$
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time [1]_, where $I(n-1)$ is the time needed to compute the
|
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inverse Laplacian. For a full matrix this is $O(n^3)$ but using
|
|
sparse methods you can achieve $O(nm{\sqrt k})$ where $k$ is the
|
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Laplacian matrix condition number.
|
|
|
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The space required is $O(nw)$ where $w$ is the width of the sparse
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Laplacian matrix. Worse case is $w=n$ for $O(n^2)$.
|
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|
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If the edges have a 'weight' attribute they will be used as
|
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weights in this algorithm. Unspecified weights are set to 1.
|
|
|
|
References
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----------
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.. [1] Centrality Measures Based on Current Flow.
|
|
Ulrik Brandes and Daniel Fleischer,
|
|
Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05).
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LNCS 3404, pp. 533-544. Springer-Verlag, 2005.
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http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf
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.. [2] A measure of betweenness centrality based on random walks,
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M. E. J. Newman, Social Networks 27, 39-54 (2005).
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"""
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try:
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import numpy as np
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except ImportError as e:
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raise ImportError(
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"current_flow_betweenness_centrality requires NumPy " "http://numpy.org/"
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) from e
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if not nx.is_connected(G):
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raise nx.NetworkXError("Graph not connected.")
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n = G.number_of_nodes()
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ordering = list(reverse_cuthill_mckee_ordering(G))
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# make a copy with integer labels according to rcm ordering
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# this could be done without a copy if we really wanted to
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mapping = dict(zip(ordering, range(n)))
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H = nx.relabel_nodes(G, mapping)
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edges = (tuple(sorted((u, v))) for u, v in H.edges())
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betweenness = dict.fromkeys(edges, 0.0)
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if normalized:
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nb = (n - 1.0) * (n - 2.0) # normalization factor
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else:
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nb = 2.0
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for row, (e) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver):
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for ss in sources:
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i = mapping[ss]
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for tt in targets:
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j = mapping[tt]
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betweenness[e] += 0.5 * np.abs(row[i] - row[j])
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betweenness[e] /= nb
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return {(ordering[s], ordering[t]): v for (s, t), v in betweenness.items()}
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