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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |

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"""Current-flow betweenness centrality measures.""" import networkx as nx from networkx.algorithms.centrality.flow_matrix import ( CGInverseLaplacian, flow_matrix_row, FullInverseLaplacian, laplacian_sparse_matrix, SuperLUInverseLaplacian, ) from networkx.utils import ( not_implemented_for, reverse_cuthill_mckee_ordering, py_random_state, ) __all__ = [ "current_flow_betweenness_centrality", "approximate_current_flow_betweenness_centrality", "edge_current_flow_betweenness_centrality", ] @py_random_state(7) @not_implemented_for("directed") def approximate_current_flow_betweenness_centrality( G, normalized=True, weight=None, dtype=float, solver="full", epsilon=0.5, kmax=10000, seed=None, ): r"""Compute the approximate current-flow betweenness centrality for nodes. Approximates the current-flow betweenness centrality within absolute error of epsilon with high probability [1]_. Parameters ---------- G : graph A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number of nodes in G. weight : string or None, optional (default=None) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype : data type (float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver : string (default='full') Type of linear solver to use for computing the flow matrix. Options are "full" (uses most memory), "lu" (recommended), and "cg" (uses least memory). epsilon: float Absolute error tolerance. kmax: int Maximum number of sample node pairs to use for approximation. seed : integer, random_state, or None (default) Indicator of random number generation state. See :ref:`Randomness<randomness>`. Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- current_flow_betweenness_centrality Notes ----- The running time is $O((1/\epsilon^2)m{\sqrt k} \log n)$ and the space required is $O(m)$ for $n$ nodes and $m$ edges. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. References ---------- .. [1] Ulrik Brandes and Daniel Fleischer: Centrality Measures Based on Current Flow. Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). LNCS 3404, pp. 533-544. Springer-Verlag, 2005. http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf """ try: import numpy as np except ImportError as e: raise ImportError( "current_flow_betweenness_centrality requires NumPy " "http://numpy.org/" ) from e if not nx.is_connected(G): raise nx.NetworkXError("Graph not connected.") solvername = { "full": FullInverseLaplacian, "lu": SuperLUInverseLaplacian, "cg": CGInverseLaplacian, } n = G.number_of_nodes() ordering = list(reverse_cuthill_mckee_ordering(G)) # make a copy with integer labels according to rcm ordering # this could be done without a copy if we really wanted to H = nx.relabel_nodes(G, dict(zip(ordering, range(n)))) L = laplacian_sparse_matrix( H, nodelist=range(n), weight=weight, dtype=dtype, format="csc" ) C = solvername[solver](L, dtype=dtype) # initialize solver betweenness = dict.fromkeys(H, 0.0) nb = (n - 1.0) * (n - 2.0) # normalization factor cstar = n * (n - 1) / nb l = 1 # parameter in approximation, adjustable k = l * int(np.ceil((cstar / epsilon) ** 2 * np.log(n))) if k > kmax: msg = f"Number random pairs k>kmax ({k}>{kmax}) " raise nx.NetworkXError(msg, "Increase kmax or epsilon") cstar2k = cstar / (2 * k) for i in range(k): s, t = seed.sample(range(n), 2) b = np.zeros(n, dtype=dtype) b[s] = 1 b[t] = -1 p = C.solve(b) for v in H: if v == s or v == t: continue for nbr in H[v]: w = H[v][nbr].get(weight, 1.0) betweenness[v] += w * np.abs(p[v] - p[nbr]) * cstar2k if normalized: factor = 1.0 else: factor = nb / 2.0 # remap to original node names and "unnormalize" if required return {ordering[k]: float(v * factor) for k, v in betweenness.items()} @not_implemented_for("directed") def current_flow_betweenness_centrality( G, normalized=True, weight=None, dtype=float, solver="full" ): r"""Compute current-flow betweenness centrality for nodes. Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths. Current-flow betweenness centrality is also known as random-walk betweenness centrality [2]_. Parameters ---------- G : graph A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number of nodes in G. weight : string or None, optional (default=None) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype : data type (float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver : string (default='full') Type of linear solver to use for computing the flow matrix. Options are "full" (uses most memory), "lu" (recommended), and "cg" (uses least memory). Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- approximate_current_flow_betweenness_centrality betweenness_centrality edge_betweenness_centrality edge_current_flow_betweenness_centrality Notes ----- Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$ time [1]_, where $I(n-1)$ is the time needed to compute the 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 Laplacian matrix condition number. The space required is $O(nw)$ where $w$ is the width of the sparse Laplacian matrix. Worse case is $w=n$ for $O(n^2)$. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. References ---------- .. [1] Centrality Measures Based on Current Flow. Ulrik Brandes and Daniel Fleischer, Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). LNCS 3404, pp. 533-544. Springer-Verlag, 2005. http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf .. [2] A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). """ if not nx.is_connected(G): raise nx.NetworkXError("Graph not connected.") n = G.number_of_nodes() ordering = list(reverse_cuthill_mckee_ordering(G)) # make a copy with integer labels according to rcm ordering # this could be done without a copy if we really wanted to H = nx.relabel_nodes(G, dict(zip(ordering, range(n)))) betweenness = dict.fromkeys(H, 0.0) # b[v]=0 for v in H for row, (s, t) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver): pos = dict(zip(row.argsort()[::-1], range(n))) for i in range(n): betweenness[s] += (i - pos[i]) * row[i] betweenness[t] += (n - i - 1 - pos[i]) * row[i] if normalized: nb = (n - 1.0) * (n - 2.0) # normalization factor else: nb = 2.0 for v in H: betweenness[v] = float((betweenness[v] - v) * 2.0 / nb) return {ordering[k]: v for k, v in betweenness.items()} @not_implemented_for("directed") def edge_current_flow_betweenness_centrality( G, normalized=True, weight=None, dtype=float, solver="full" ): r"""Compute current-flow betweenness centrality for edges. Current-flow betweenness centrality uses an electrical current model for information spreading in contrast to betweenness centrality which uses shortest paths. Current-flow betweenness centrality is also known as random-walk betweenness centrality [2]_. Parameters ---------- G : graph A NetworkX graph normalized : bool, optional (default=True) If True the betweenness values are normalized by 2/[(n-1)(n-2)] where n is the number of nodes in G. weight : string or None, optional (default=None) Key for edge data used as the edge weight. If None, then use 1 as each edge weight. dtype : data type (default=float) Default data type for internal matrices. Set to np.float32 for lower memory consumption. solver : string (default='full') Type of linear solver to use for computing the flow matrix. Options are "full" (uses most memory), "lu" (recommended), and "cg" (uses least memory). Returns ------- nodes : dictionary Dictionary of edge tuples with betweenness centrality as the value. Raises ------ NetworkXError The algorithm does not support DiGraphs. If the input graph is an instance of DiGraph class, NetworkXError is raised. See Also -------- betweenness_centrality edge_betweenness_centrality current_flow_betweenness_centrality Notes ----- Current-flow betweenness can be computed in $O(I(n-1)+mn \log n)$ time [1]_, where $I(n-1)$ is the time needed to compute the 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 Laplacian matrix condition number. The space required is $O(nw)$ where $w$ is the width of the sparse Laplacian matrix. Worse case is $w=n$ for $O(n^2)$. If the edges have a 'weight' attribute they will be used as weights in this algorithm. Unspecified weights are set to 1. References ---------- .. [1] Centrality Measures Based on Current Flow. Ulrik Brandes and Daniel Fleischer, Proc. 22nd Symp. Theoretical Aspects of Computer Science (STACS '05). LNCS 3404, pp. 533-544. Springer-Verlag, 2005. http://algo.uni-konstanz.de/publications/bf-cmbcf-05.pdf .. [2] A measure of betweenness centrality based on random walks, M. E. J. Newman, Social Networks 27, 39-54 (2005). """ from networkx.utils import reverse_cuthill_mckee_ordering if not nx.is_connected(G): raise nx.NetworkXError("Graph not connected.") n = G.number_of_nodes() ordering = list(reverse_cuthill_mckee_ordering(G)) # make a copy with integer labels according to rcm ordering # this could be done without a copy if we really wanted to H = nx.relabel_nodes(G, dict(zip(ordering, range(n)))) edges = (tuple(sorted((u, v))) for u, v in H.edges()) betweenness = dict.fromkeys(edges, 0.0) if normalized: nb = (n - 1.0) * (n - 2.0) # normalization factor else: nb = 2.0 for row, (e) in flow_matrix_row(H, weight=weight, dtype=dtype, solver=solver): pos = dict(zip(row.argsort()[::-1], range(1, n + 1))) for i in range(n): betweenness[e] += (i + 1 - pos[i]) * row[i] betweenness[e] += (n - i - pos[i]) * row[i] betweenness[e] /= nb return {(ordering[s], ordering[t]): float(v) for (s, t), v in betweenness.items()}