## Mercurial > repos > shellac > sam_consensus_v3

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

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"""Betweenness centrality measures for subsets of nodes.""" import warnings from networkx.algorithms.centrality.betweenness import ( _single_source_dijkstra_path_basic as dijkstra, ) from networkx.algorithms.centrality.betweenness import ( _single_source_shortest_path_basic as shortest_path, ) __all__ = [ "betweenness_centrality_subset", "betweenness_centrality_source", "edge_betweenness_centrality_subset", ] def betweenness_centrality_subset(G, sources, targets, normalized=False, weight=None): r"""Compute betweenness centrality for a subset of nodes. .. math:: c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)} where $S$ is the set of sources, $T$ is the set of targets, $\sigma(s, t)$ is the number of shortest $(s, t)$-paths, and $\sigma(s, t|v)$ is the number of those paths passing through some node $v$ other than $s, t$. If $s = t$, $\sigma(s, t) = 1$, and if $v \in {s, t}$, $\sigma(s, t|v) = 0$ [2]_. Parameters ---------- G : graph A NetworkX graph. sources: list of nodes Nodes to use as sources for shortest paths in betweenness targets: list of nodes Nodes to use as targets for shortest paths in betweenness normalized : bool, optional If True the betweenness values are normalized by $2/((n-1)(n-2))$ for graphs, and $1/((n-1)(n-2))$ for directed graphs where $n$ is the number of nodes in G. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns ------- nodes : dictionary Dictionary of nodes with betweenness centrality as the value. See Also -------- edge_betweenness_centrality load_centrality Notes ----- The basic algorithm is from [1]_. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. The normalization might seem a little strange but it is designed to make betweenness_centrality(G) be the same as betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()). The total number of paths between source and target is counted differently for directed and undirected graphs. Directed paths are easy to count. Undirected paths are tricky: should a path from "u" to "v" count as 1 undirected path or as 2 directed paths? For betweenness_centrality we report the number of undirected paths when G is undirected. For betweenness_centrality_subset the reporting is different. If the source and target subsets are the same, then we want to count undirected paths. But if the source and target subsets differ -- for example, if sources is {0} and targets is {1}, then we are only counting the paths in one direction. They are undirected paths but we are counting them in a directed way. To count them as undirected paths, each should count as half a path. References ---------- .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf """ b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G for s in sources: # single source shortest paths if weight is None: # use BFS S, P, sigma = shortest_path(G, s) else: # use Dijkstra's algorithm S, P, sigma = dijkstra(G, s, weight) b = _accumulate_subset(b, S, P, sigma, s, targets) b = _rescale(b, len(G), normalized=normalized, directed=G.is_directed()) return b def edge_betweenness_centrality_subset( G, sources, targets, normalized=False, weight=None ): r"""Compute betweenness centrality for edges for a subset of nodes. .. math:: c_B(v) =\sum_{s\in S,t \in T} \frac{\sigma(s, t|e)}{\sigma(s, t)} where $S$ is the set of sources, $T$ is the set of targets, $\sigma(s, t)$ is the number of shortest $(s, t)$-paths, and $\sigma(s, t|e)$ is the number of those paths passing through edge $e$ [2]_. Parameters ---------- G : graph A networkx graph. sources: list of nodes Nodes to use as sources for shortest paths in betweenness targets: list of nodes Nodes to use as targets for shortest paths in betweenness normalized : bool, optional If True the betweenness values are normalized by `2/(n(n-1))` for graphs, and `1/(n(n-1))` for directed graphs where `n` is the number of nodes in G. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. Returns ------- edges : dictionary Dictionary of edges with Betweenness centrality as the value. See Also -------- betweenness_centrality edge_load Notes ----- The basic algorithm is from [1]_. For weighted graphs the edge weights must be greater than zero. Zero edge weights can produce an infinite number of equal length paths between pairs of nodes. The normalization might seem a little strange but it is the same as in edge_betweenness_centrality() and is designed to make edge_betweenness_centrality(G) be the same as edge_betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()). References ---------- .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://www.inf.uni-konstanz.de/algo/publications/b-vspbc-08.pdf """ b = dict.fromkeys(G, 0.0) # b[v]=0 for v in G b.update(dict.fromkeys(G.edges(), 0.0)) # b[e] for e in G.edges() for s in sources: # single source shortest paths if weight is None: # use BFS S, P, sigma = shortest_path(G, s) else: # use Dijkstra's algorithm S, P, sigma = dijkstra(G, s, weight) b = _accumulate_edges_subset(b, S, P, sigma, s, targets) for n in G: # remove nodes to only return edges del b[n] b = _rescale_e(b, len(G), normalized=normalized, directed=G.is_directed()) return b # obsolete name def betweenness_centrality_source(G, normalized=True, weight=None, sources=None): msg = "betweenness_centrality_source --> betweenness_centrality_subset" warnings.warn(msg, DeprecationWarning) if sources is None: sources = G.nodes() targets = list(G) return betweenness_centrality_subset(G, sources, targets, normalized, weight) def _accumulate_subset(betweenness, S, P, sigma, s, targets): delta = dict.fromkeys(S, 0.0) target_set = set(targets) - {s} while S: w = S.pop() if w in target_set: coeff = (delta[w] + 1.0) / sigma[w] else: coeff = delta[w] / sigma[w] for v in P[w]: delta[v] += sigma[v] * coeff if w != s: betweenness[w] += delta[w] return betweenness def _accumulate_edges_subset(betweenness, S, P, sigma, s, targets): """edge_betweenness_centrality_subset helper.""" delta = dict.fromkeys(S, 0) target_set = set(targets) while S: w = S.pop() for v in P[w]: if w in target_set: c = (sigma[v] / sigma[w]) * (1.0 + delta[w]) else: c = delta[w] / len(P[w]) if (v, w) not in betweenness: betweenness[(w, v)] += c else: betweenness[(v, w)] += c delta[v] += c if w != s: betweenness[w] += delta[w] return betweenness def _rescale(betweenness, n, normalized, directed=False): """betweenness_centrality_subset helper.""" if normalized: if n <= 2: scale = None # no normalization b=0 for all nodes else: scale = 1.0 / ((n - 1) * (n - 2)) else: # rescale by 2 for undirected graphs if not directed: scale = 0.5 else: scale = None if scale is not None: for v in betweenness: betweenness[v] *= scale return betweenness def _rescale_e(betweenness, n, normalized, directed=False): """edge_betweenness_centrality_subset helper.""" if normalized: if n <= 1: scale = None # no normalization b=0 for all nodes else: scale = 1.0 / (n * (n - 1)) else: # rescale by 2 for undirected graphs if not directed: scale = 0.5 else: scale = None if scale is not None: for v in betweenness: betweenness[v] *= scale return betweenness