Source code for networkx.algorithms.bipartite.centrality

import networkx as nx

__all__ = ["degree_centrality", "betweenness_centrality", "closeness_centrality"]

[docs]def degree_centrality(G, nodes): r"""Compute the degree centrality for nodes in a bipartite network. The degree centrality for a node v is the fraction of nodes connected to it. Parameters ---------- G : graph A bipartite network nodes : list or container Container with all nodes in one bipartite node set. Returns ------- centrality : dictionary Dictionary keyed by node with bipartite degree centrality as the value. See Also -------- betweenness_centrality closeness_centrality :func:~networkx.algorithms.bipartite.basic.sets :func:~networkx.algorithms.bipartite.basic.is_bipartite Notes ----- The nodes input parameter must contain all nodes in one bipartite node set, but the dictionary returned contains all nodes from both bipartite node sets. See :mod:bipartite documentation <networkx.algorithms.bipartite> for further details on how bipartite graphs are handled in NetworkX. For unipartite networks, the degree centrality values are normalized by dividing by the maximum possible degree (which is n-1 where n is the number of nodes in G). In the bipartite case, the maximum possible degree of a node in a bipartite node set is the number of nodes in the opposite node set _. The degree centrality for a node v in the bipartite sets U with n nodes and V with m nodes is .. math:: d_{v} = \frac{deg(v)}{m}, \mbox{for} v \in U , d_{v} = \frac{deg(v)}{n}, \mbox{for} v \in V , where deg(v) is the degree of node v. References ---------- ..  Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook of Social Network Analysis. Sage Publications. https://dx.doi.org/10.4135/9781446294413.n28 """ top = set(nodes) bottom = set(G) - top s = 1.0 / len(bottom) centrality = {n: d * s for n, d in G.degree(top)} s = 1.0 / len(top) centrality.update({n: d * s for n, d in G.degree(bottom)}) return centrality
[docs]def betweenness_centrality(G, nodes): r"""Compute betweenness centrality for nodes in a bipartite network. Betweenness centrality of a node v is the sum of the fraction of all-pairs shortest paths that pass through v. Values of betweenness are normalized by the maximum possible value which for bipartite graphs is limited by the relative size of the two node sets _. Let n be the number of nodes in the node set U and m be the number of nodes in the node set V, then nodes in U are normalized by dividing by .. math:: \frac{1}{2} [m^2 (s + 1)^2 + m (s + 1)(2t - s - 1) - t (2s - t + 3)] , where .. math:: s = (n - 1) \div m , t = (n - 1) \mod m , and nodes in V are normalized by dividing by .. math:: \frac{1}{2} [n^2 (p + 1)^2 + n (p + 1)(2r - p - 1) - r (2p - r + 3)] , where, .. math:: p = (m - 1) \div n , r = (m - 1) \mod n . Parameters ---------- G : graph A bipartite graph nodes : list or container Container with all nodes in one bipartite node set. Returns ------- betweenness : dictionary Dictionary keyed by node with bipartite betweenness centrality as the value. See Also -------- degree_centrality closeness_centrality :func:~networkx.algorithms.bipartite.basic.sets :func:~networkx.algorithms.bipartite.basic.is_bipartite Notes ----- The nodes input parameter must contain all nodes in one bipartite node set, but the dictionary returned contains all nodes from both node sets. See :mod:bipartite documentation <networkx.algorithms.bipartite> for further details on how bipartite graphs are handled in NetworkX. References ---------- ..  Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook of Social Network Analysis. Sage Publications. https://dx.doi.org/10.4135/9781446294413.n28 """ top = set(nodes) bottom = set(G) - top n = float(len(top)) m = float(len(bottom)) s = (n - 1) // m t = (n - 1) % m bet_max_top = ( ((m ** 2) * ((s + 1) ** 2)) + (m * (s + 1) * (2 * t - s - 1)) - (t * ((2 * s) - t + 3)) ) / 2.0 p = (m - 1) // n r = (m - 1) % n bet_max_bot = ( ((n ** 2) * ((p + 1) ** 2)) + (n * (p + 1) * (2 * r - p - 1)) - (r * ((2 * p) - r + 3)) ) / 2.0 betweenness = nx.betweenness_centrality(G, normalized=False, weight=None) for node in top: betweenness[node] /= bet_max_top for node in bottom: betweenness[node] /= bet_max_bot return betweenness
[docs]def closeness_centrality(G, nodes, normalized=True): r"""Compute the closeness centrality for nodes in a bipartite network. The closeness of a node is the distance to all other nodes in the graph or in the case that the graph is not connected to all other nodes in the connected component containing that node. Parameters ---------- G : graph A bipartite network nodes : list or container Container with all nodes in one bipartite node set. normalized : bool, optional If True (default) normalize by connected component size. Returns ------- closeness : dictionary Dictionary keyed by node with bipartite closeness centrality as the value. See Also -------- betweenness_centrality degree_centrality :func:~networkx.algorithms.bipartite.basic.sets :func:~networkx.algorithms.bipartite.basic.is_bipartite Notes ----- The nodes input parameter must contain all nodes in one bipartite node set, but the dictionary returned contains all nodes from both node sets. See :mod:bipartite documentation <networkx.algorithms.bipartite> for further details on how bipartite graphs are handled in NetworkX. Closeness centrality is normalized by the minimum distance possible. In the bipartite case the minimum distance for a node in one bipartite node set is 1 from all nodes in the other node set and 2 from all other nodes in its own set _. Thus the closeness centrality for node v in the two bipartite sets U with n nodes and V with m nodes is .. math:: c_{v} = \frac{m + 2(n - 1)}{d}, \mbox{for} v \in U, c_{v} = \frac{n + 2(m - 1)}{d}, \mbox{for} v \in V, where d is the sum of the distances from v to all other nodes. Higher values of closeness indicate higher centrality. As in the unipartite case, setting normalized=True causes the values to normalized further to n-1 / size(G)-1 where n is the number of nodes in the connected part of graph containing the node. If the graph is not completely connected, this algorithm computes the closeness centrality for each connected part separately. References ---------- ..  Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook of Social Network Analysis. Sage Publications. https://dx.doi.org/10.4135/9781446294413.n28 """ closeness = {} path_length = nx.single_source_shortest_path_length top = set(nodes) bottom = set(G) - top n = float(len(top)) m = float(len(bottom)) for node in top: sp = dict(path_length(G, node)) totsp = sum(sp.values()) if totsp > 0.0 and len(G) > 1: closeness[node] = (m + 2 * (n - 1)) / totsp if normalized: s = (len(sp) - 1.0) / (len(G) - 1) closeness[node] *= s else: closeness[n] = 0.0 for node in bottom: sp = dict(path_length(G, node)) totsp = sum(sp.values()) if totsp > 0.0 and len(G) > 1: closeness[node] = (n + 2 * (m - 1)) / totsp if normalized: s = (len(sp) - 1.0) / (len(G) - 1) closeness[node] *= s else: closeness[n] = 0.0 return closeness