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 the same as in betweenness_centrality() and is designed to make betweenness_centrality(G) be the same as 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 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, :math:`\sigma(s, t)` is the number of shortest `(s, t)`-paths, and :math:`\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
def edge_betweenness_centrality_subset(G,sources,targets, normalized=False, weight=None): """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 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
def percolation_centrality(G, attribute="percolation", states=None, weight=None): r"""Compute the percolation centrality for nodes. Percolation centrality of a node $v$, at a given time, is defined as the proportion of ‘percolated paths’ that go through that node. This measure quantifies relative impact of nodes based on their topological connectivity, as well as their percolation states. Percolation states of nodes are used to depict network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of disease over a network of towns) over time. In this measure usually the percolation state is expressed as a decimal between 0.0 and 1.0. When all nodes are in the same percolated state this measure is equivalent to betweenness centrality. Parameters ---------- G : graph A NetworkX graph. attribute : None or string, optional (default='percolation') Name of the node attribute to use for percolation state, used if `states` is None. states : None or dict, optional (default=None) Specify percolation states for the nodes, nodes as keys states as values. 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 percolation centrality as the value. See Also -------- betweenness_centrality Notes ----- The algorithm is from Mahendra Piraveenan, Mikhail Prokopenko, and Liaquat Hossain [1]_ Pair dependecies are calculated and accumulated using [2]_ 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] Mahendra Piraveenan, Mikhail Prokopenko, Liaquat Hossain Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0053095 .. [2] 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 """ percolation = dict.fromkeys(G, 0.0) # b[v]=0 for v in G nodes = G if states is None: states = nx.get_node_attributes(nodes, attribute) # sum of all percolation states p_sigma_x_t = 0.0 for v in states.values(): p_sigma_x_t += v for s in nodes: # 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) # accumulation percolation = _accumulate_percolation(percolation, G, S, P, sigma, s, states, p_sigma_x_t) n = len(G) for v in percolation: percolation[v] *= 1 / (n - 2) return percolation
def betweenness_centrality_subset(G,sources,targets, normalized=False, weighted_edges=False): """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 :math:`S` is the set of sources, :math:`T` is the set of targets, :math:`\\sigma(s, t)` is the number of shortest :math:`(s, t)`-paths, and :math:`\\sigma(s, t|v)` is the number of those paths passing through some node :math:`v` other than :math:`s, t`. If :math:`s = t`, :math:`\\sigma(s, t) = 1`, and if :math:`v \\in {s, t}`, :math:`\\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 :math:`1/(n-1)(n-2)` where :math:`n` is the number of nodes in G. weighted_edges : bool, optional Consider the edge weights in determining the shortest paths. The edge weights must be greater than zero. If False, all edge weights are considered equal. 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 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. The normalization might seem a little strange but it is the same as in betweenness_centrality() and is designed to make betweenness_centrality(G) be the same as betweenness_centrality_subset(G,sources=G.nodes(),targets=G.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 """ b=dict.fromkeys(G,0.0) # b[v]=0 for v in G for s in sources: # single source shortest paths if weighted_edges: # use Dijkstra's algorithm S,P,sigma=dijkstra(G,s) else: # use BFS S,P,sigma=shortest_path(G,s) b=_accumulate_subset(b,S,P,sigma,s,targets) b=_rescale(b,normalized=normalized,directed=G.is_directed()) return b
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. Weights are used to calculate weighted shortest paths, so they are interpreted as distances. 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. https://doi.org/10.1080/0022250X.2001.9990249 .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. https://doi.org/10.1016/j.socnet.2007.11.001 """ 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, weighted_edges=False): """Compute betweenness centrality for edges. Betweenness centrality of an edge is the fraction of all shortest paths that pass through that edge. 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 b=b/(n-1)(n-2) where n is the number of nodes in G. weighted_edges : bool, optional Consider the edge weights in determining the shortest paths. The edge weights must be greater than zero. If False, all edge weights are considered equal. Returns ------- edges : dictionary Dictionary of edges with Betweenness centrality as the value. See Also -------- betweenness_centrality edge_load Notes ----- The basic 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. 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] 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 """ 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 weighted_edges: # use Dijkstra's algorithm S,P,sigma=dijkstra(G,s) else: # use BFS S,P,sigma=shortest_path(G,s) 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(b,normalized=normalized,directed=G.is_directed()) return b
def edge_betweenness_centrality_subset(G, sources, targets, normalized=False, weighted_edges=False): """Compute betweenness centrality for edges. Betweenness centrality of an edge is the fraction of all shortest paths that pass through that edge. 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 b=b/(n-1)(n-2) where n is the number of nodes in G. weighted_edges : bool, optional Consider the edge weights in determining the shortest paths. The edge weights must be greater than zero. If False, all edge weights are considered equal. Returns ------- edges : dictionary Dictionary of edges with Betweenness centrality as the value. See Also -------- betweenness_centrality edge_load Notes ----- The basic 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. 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] 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 """ 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 weighted_edges: # use Dijkstra's algorithm S, P, sigma = dijkstra(G, s) else: # use BFS S, P, sigma = shortest_path(G, s) 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(b, normalized=normalized, directed=G.is_directed()) return b
def percolation_centrality(G, attribute='percolation', states=None, weight=None): r"""Compute the percolation centrality for nodes. Percolation centrality of a node $v$, at a given time, is defined as the proportion of ‘percolated paths’ that go through that node. This measure quantifies relative impact of nodes based on their topological connectivity, as well as their percolation states. Percolation states of nodes are used to depict network percolation scenarios (such as during infection transmission in a social network of individuals, spreading of computer viruses on computer networks, or transmission of disease over a network of towns) over time. In this measure usually the percolation state is expressed as a decimal between 0.0 and 1.0. When all nodes are in the same percolated state this measure is equivalent to betweenness centrality. Parameters ---------- G : graph A NetworkX graph. attribute : None or string, optional (default='percolation') Name of the node attribute to use for percolation state, used if `states` is None. states : None or dict, optional (default=None) Specify percolation states for the nodes, nodes as keys states as values. 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 percolation centrality as the value. See Also -------- betweenness_centrality Notes ----- The algorithm is from Mahendra Piraveenan, Mikhail Prokopenko, and Liaquat Hossain [1]_ Pair dependecies are calculated and accumulated using [2]_ 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] Mahendra Piraveenan, Mikhail Prokopenko, Liaquat Hossain Percolation Centrality: Quantifying Graph-Theoretic Impact of Nodes during Percolation in Networks http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0053095 .. [2] 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 """ percolation = dict.fromkeys(G, 0.0) # b[v]=0 for v in G nodes = G if states is None: states = nx.get_node_attributes(nodes, attribute) # sum of all percolation states p_sigma_x_t = 0.0 for v in states.values(): p_sigma_x_t += v for s in nodes: # 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) # accumulation percolation = _accumulate_percolation(percolation, G, S, P, sigma, s, states, p_sigma_x_t) n = len(G) for v in percolation: percolation[v] *= 1 / (n - 2) return percolation