Mercurial > repos > shellac > sam_consensus_v3
diff env/lib/python3.9/site-packages/networkx/algorithms/centrality/betweenness.py @ 0:4f3585e2f14b draft default tip
"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/env/lib/python3.9/site-packages/networkx/algorithms/centrality/betweenness.py Mon Mar 22 18:12:50 2021 +0000 @@ -0,0 +1,392 @@ +"""Betweenness centrality measures.""" +from heapq import heappush, heappop +from itertools import count +import warnings + +from networkx.utils import py_random_state +from networkx.utils.decorators import not_implemented_for + +__all__ = ["betweenness_centrality", "edge_betweenness_centrality", "edge_betweenness"] + + +@py_random_state(5) +@not_implemented_for("multigraph") +def betweenness_centrality( + G, k=None, normalized=True, weight=None, endpoints=False, seed=None +): + r"""Compute the shortest-path betweenness centrality for nodes. + + Betweenness centrality of a node $v$ is the sum of the + fraction of all-pairs shortest paths that pass through $v$ + + .. math:: + + c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, t|v)}{\sigma(s, t)} + + where $V$ is the set of nodes, $\sigma(s, t)$ 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. + + k : int, optional (default=None) + If k is not None use k node samples to estimate betweenness. + The value of k <= n where n is the number of nodes in the graph. + Higher values give better approximation. + + 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. + + endpoints : bool, optional + If True include the endpoints in the shortest path counts. + + seed : integer, random_state, or None (default) + Indicator of random number generation state. + See :ref:`Randomness<randomness>`. + Note that this is only used if k is not None. + + Returns + ------- + nodes : dictionary + Dictionary of nodes with betweenness centrality as the value. + + See Also + -------- + edge_betweenness_centrality + load_centrality + + Notes + ----- + The algorithm is from Ulrik Brandes [1]_. + See [4]_ for the original first published version and [2]_ for details on + algorithms for variations and related metrics. + + For approximate betweenness calculations set k=#samples to use + k nodes ("pivots") to estimate the betweenness values. For an estimate + of the number of pivots needed see [3]_. + + 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 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 + .. [3] Ulrik Brandes and Christian Pich: + Centrality Estimation in Large Networks. + International Journal of Bifurcation and Chaos 17(7):2303-2318, 2007. + http://www.inf.uni-konstanz.de/algo/publications/bp-celn-06.pdf + .. [4] Linton C. Freeman: + A set of measures of centrality based on betweenness. + Sociometry 40: 35–41, 1977 + http://moreno.ss.uci.edu/23.pdf + """ + betweenness = dict.fromkeys(G, 0.0) # b[v]=0 for v in G + if k is None: + nodes = G + else: + nodes = seed.sample(G.nodes(), k) + for s in nodes: + # single source shortest paths + if weight is None: # use BFS + S, P, sigma = _single_source_shortest_path_basic(G, s) + else: # use Dijkstra's algorithm + S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight) + # accumulation + if endpoints: + betweenness = _accumulate_endpoints(betweenness, S, P, sigma, s) + else: + betweenness = _accumulate_basic(betweenness, S, P, sigma, s) + # rescaling + betweenness = _rescale( + betweenness, + len(G), + normalized=normalized, + directed=G.is_directed(), + k=k, + endpoints=endpoints, + ) + return betweenness + + +@py_random_state(4) +def edge_betweenness_centrality(G, k=None, normalized=True, weight=None, seed=None): + r"""Compute betweenness centrality for edges. + + Betweenness centrality of an edge $e$ is the sum of the + fraction of all-pairs shortest paths that pass through $e$ + + .. math:: + + c_B(e) =\sum_{s,t \in V} \frac{\sigma(s, t|e)}{\sigma(s, t)} + + where $V$ is the set of nodes, $\sigma(s, t)$ 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. + + k : int, optional (default=None) + If k is not None use k node samples to estimate betweenness. + The value of k <= n where n is the number of nodes in the graph. + Higher values give better approximation. + + 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. + + seed : integer, random_state, or None (default) + Indicator of random number generation state. + See :ref:`Randomness<randomness>`. + Note that this is only used if k is not None. + + Returns + ------- + edges : dictionary + Dictionary of edges with betweenness centrality as the value. + + See Also + -------- + betweenness_centrality + edge_load + + Notes + ----- + The algorithm is from Ulrik Brandes [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. + + References + ---------- + .. [1] A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, + 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 + """ + betweenness = dict.fromkeys(G, 0.0) # b[v]=0 for v in G + # b[e]=0 for e in G.edges() + betweenness.update(dict.fromkeys(G.edges(), 0.0)) + if k is None: + nodes = G + else: + nodes = seed.sample(G.nodes(), k) + for s in nodes: + # single source shortest paths + if weight is None: # use BFS + S, P, sigma = _single_source_shortest_path_basic(G, s) + else: # use Dijkstra's algorithm + S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight) + # accumulation + betweenness = _accumulate_edges(betweenness, S, P, sigma, s) + # rescaling + for n in G: # remove nodes to only return edges + del betweenness[n] + betweenness = _rescale_e( + betweenness, len(G), normalized=normalized, directed=G.is_directed() + ) + return betweenness + + +# obsolete name +def edge_betweenness(G, k=None, normalized=True, weight=None, seed=None): + warnings.warn( + "edge_betweeness is replaced by edge_betweenness_centrality", DeprecationWarning + ) + return edge_betweenness_centrality(G, k, normalized, weight, seed) + + +# helpers for betweenness centrality + + +def _single_source_shortest_path_basic(G, s): + S = [] + P = {} + for v in G: + P[v] = [] + sigma = dict.fromkeys(G, 0.0) # sigma[v]=0 for v in G + D = {} + sigma[s] = 1.0 + D[s] = 0 + Q = [s] + while Q: # use BFS to find shortest paths + v = Q.pop(0) + S.append(v) + Dv = D[v] + sigmav = sigma[v] + for w in G[v]: + if w not in D: + Q.append(w) + D[w] = Dv + 1 + if D[w] == Dv + 1: # this is a shortest path, count paths + sigma[w] += sigmav + P[w].append(v) # predecessors + return S, P, sigma + + +def _single_source_dijkstra_path_basic(G, s, weight): + # modified from Eppstein + S = [] + P = {} + for v in G: + P[v] = [] + sigma = dict.fromkeys(G, 0.0) # sigma[v]=0 for v in G + D = {} + sigma[s] = 1.0 + push = heappush + pop = heappop + seen = {s: 0} + c = count() + Q = [] # use Q as heap with (distance,node id) tuples + push(Q, (0, next(c), s, s)) + while Q: + (dist, _, pred, v) = pop(Q) + if v in D: + continue # already searched this node. + sigma[v] += sigma[pred] # count paths + S.append(v) + D[v] = dist + for w, edgedata in G[v].items(): + vw_dist = dist + edgedata.get(weight, 1) + if w not in D and (w not in seen or vw_dist < seen[w]): + seen[w] = vw_dist + push(Q, (vw_dist, next(c), v, w)) + sigma[w] = 0.0 + P[w] = [v] + elif vw_dist == seen[w]: # handle equal paths + sigma[w] += sigma[v] + P[w].append(v) + return S, P, sigma + + +def _accumulate_basic(betweenness, S, P, sigma, s): + delta = dict.fromkeys(S, 0) + while S: + w = S.pop() + coeff = (1 + 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_endpoints(betweenness, S, P, sigma, s): + betweenness[s] += len(S) - 1 + delta = dict.fromkeys(S, 0) + while S: + w = S.pop() + coeff = (1 + delta[w]) / sigma[w] + for v in P[w]: + delta[v] += sigma[v] * coeff + if w != s: + betweenness[w] += delta[w] + 1 + return betweenness + + +def _accumulate_edges(betweenness, S, P, sigma, s): + delta = dict.fromkeys(S, 0) + while S: + w = S.pop() + coeff = (1 + delta[w]) / sigma[w] + for v in P[w]: + c = sigma[v] * coeff + 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, k=None, endpoints=False): + if normalized: + if endpoints: + if n < 2: + scale = None # no normalization + else: + # Scale factor should include endpoint nodes + scale = 1 / (n * (n - 1)) + elif n <= 2: + scale = None # no normalization b=0 for all nodes + else: + scale = 1 / ((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: + if k is not None: + scale = scale * n / k + for v in betweenness: + betweenness[v] *= scale + return betweenness + + +def _rescale_e(betweenness, n, normalized, directed=False, k=None): + if normalized: + if n <= 1: + scale = None # no normalization b=0 for all nodes + else: + scale = 1 / (n * (n - 1)) + else: # rescale by 2 for undirected graphs + if not directed: + scale = 0.5 + else: + scale = None + if scale is not None: + if k is not None: + scale = scale * n / k + for v in betweenness: + betweenness[v] *= scale + return betweenness