# Source code for networkx.algorithms.centrality.betweenness

"""Betweenness centrality measures."""
from heapq import heappush, heappop
from itertools import count

from networkx.utils import py_random_state
from networkx.utils.decorators import not_implemented_for

__all__ = ['betweenness_centrality', 'edge_betweenness_centrality',
'edge_betweenness']

[docs]@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
[docs]@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): 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