"""Group centrality measures.""" from copy import deepcopy import networkx as nx from networkx.algorithms.centrality.betweenness import ( _accumulate_endpoints, _single_source_dijkstra_path_basic, _single_source_shortest_path_basic, ) from networkx.utils.decorators import not_implemented_for __all__ = [ "group_betweenness_centrality", "group_closeness_centrality", "group_degree_centrality", "group_in_degree_centrality", "group_out_degree_centrality", "prominent_group", ] @nx._dispatchable(edge_attrs="weight") def group_betweenness_centrality(G, C, normalized=True, weight=None, endpoints=False): r"""Compute the group betweenness centrality for a group of nodes. Group betweenness centrality of a group of nodes $C$ is the sum of the fraction of all-pairs shortest paths that pass through any vertex in $C$ .. 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 $\sigma(s, t|C)$ is the number of those paths passing through some node in group $C$. Note that $(s, t)$ are not members of the group ($V-C$ is the set of nodes in $V$ that are not in $C$). Parameters ---------- G : graph A NetworkX graph. C : list or set or list of lists or list of sets A group or a list of groups containing nodes which belong to G, for which group betweenness centrality is to be calculated. normalized : bool, optional (default=True) If True, group betweenness is normalized by `1/((|V|-|C|)(|V|-|C|-1))` where `|V|` is the number of nodes in G and `|C|` is the number of nodes in C. 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. The weight of an edge is treated as the length or distance between the two sides. endpoints : bool, optional (default=False) If True include the endpoints in the shortest path counts. Raises ------ NodeNotFound If node(s) in C are not present in G. Returns ------- betweenness : list of floats or float If C is a single group then return a float. If C is a list with several groups then return a list of group betweenness centralities. See Also -------- betweenness_centrality Notes ----- Group betweenness centrality is described in [1]_ and its importance discussed in [3]_. The initial implementation of the algorithm is mentioned in [2]_. This function uses an improved algorithm presented in [4]_. The number of nodes in the group must be a maximum of n - 2 where `n` is the total number of nodes in the graph. 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 between "u" and "v" are counted as two possible paths (one each direction) while undirected paths between "u" and "v" are counted as one path. Said another way, the sum in the expression above is over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs. References ---------- .. [1] M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness Centrality and their Generic Computation. Social Networks 30(2):136-145, 2008. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf .. [3] Sourav Medya et. al.: Group Centrality Maximization via Network Design. SIAM International Conference on Data Mining, SDM 2018, 126–134. https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev. "Fast algorithm for successive computation of group betweenness centrality." https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709 """ GBC = [] # initialize betweenness list_of_groups = True # check weather C contains one or many groups if any(el in G for el in C): C = [C] list_of_groups = False set_v = {node for group in C for node in group} if set_v - G.nodes: # element(s) of C not in G raise nx.NodeNotFound(f"The node(s) {set_v - G.nodes} are in C but not in G.") # pre-processing PB, sigma, D = _group_preprocessing(G, set_v, weight) # the algorithm for each group for group in C: group = set(group) # set of nodes in group # initialize the matrices of the sigma and the PB GBC_group = 0 sigma_m = deepcopy(sigma) PB_m = deepcopy(PB) sigma_m_v = deepcopy(sigma_m) PB_m_v = deepcopy(PB_m) for v in group: GBC_group += PB_m[v][v] for x in group: for y in group: dxvy = 0 dxyv = 0 dvxy = 0 if not ( sigma_m[x][y] == 0 or sigma_m[x][v] == 0 or sigma_m[v][y] == 0 ): if D[x][v] == D[x][y] + D[y][v]: dxyv = sigma_m[x][y] * sigma_m[y][v] / sigma_m[x][v] if D[x][y] == D[x][v] + D[v][y]: dxvy = sigma_m[x][v] * sigma_m[v][y] / sigma_m[x][y] if D[v][y] == D[v][x] + D[x][y]: dvxy = sigma_m[v][x] * sigma[x][y] / sigma[v][y] sigma_m_v[x][y] = sigma_m[x][y] * (1 - dxvy) PB_m_v[x][y] = PB_m[x][y] - PB_m[x][y] * dxvy if y != v: PB_m_v[x][y] -= PB_m[x][v] * dxyv if x != v: PB_m_v[x][y] -= PB_m[v][y] * dvxy sigma_m, sigma_m_v = sigma_m_v, sigma_m PB_m, PB_m_v = PB_m_v, PB_m # endpoints v, c = len(G), len(group) if not endpoints: scale = 0 # if the graph is connected then subtract the endpoints from # the count for all the nodes in the graph. else count how many # nodes are connected to the group's nodes and subtract that. if nx.is_directed(G): if nx.is_strongly_connected(G): scale = c * (2 * v - c - 1) elif nx.is_connected(G): scale = c * (2 * v - c - 1) if scale == 0: for group_node1 in group: for node in D[group_node1]: if node != group_node1: if node in group: scale += 1 else: scale += 2 GBC_group -= scale # normalized if normalized: scale = 1 / ((v - c) * (v - c - 1)) GBC_group *= scale # If undirected than count only the undirected edges elif not G.is_directed(): GBC_group /= 2 GBC.append(GBC_group) if list_of_groups: return GBC return GBC[0] def _group_preprocessing(G, set_v, weight): sigma = {} delta = {} D = {} betweenness = dict.fromkeys(G, 0) for s in G: if weight is None: # use BFS S, P, sigma[s], D[s] = _single_source_shortest_path_basic(G, s) else: # use Dijkstra's algorithm S, P, sigma[s], D[s] = _single_source_dijkstra_path_basic(G, s, weight) betweenness, delta[s] = _accumulate_endpoints(betweenness, S, P, sigma[s], s) for i in delta[s]: # add the paths from s to i and rescale sigma if s != i: delta[s][i] += 1 if weight is not None: sigma[s][i] = sigma[s][i] / 2 # building the path betweenness matrix only for nodes that appear in the group PB = dict.fromkeys(G) for group_node1 in set_v: PB[group_node1] = dict.fromkeys(G, 0.0) for group_node2 in set_v: if group_node2 not in D[group_node1]: continue for node in G: # if node is connected to the two group nodes than continue if group_node2 in D[node] and group_node1 in D[node]: if ( D[node][group_node2] == D[node][group_node1] + D[group_node1][group_node2] ): PB[group_node1][group_node2] += ( delta[node][group_node2] * sigma[node][group_node1] * sigma[group_node1][group_node2] / sigma[node][group_node2] ) return PB, sigma, D @nx._dispatchable(edge_attrs="weight") def prominent_group( G, k, weight=None, C=None, endpoints=False, normalized=True, greedy=False ): r"""Find the prominent group of size $k$ in graph $G$. The prominence of the group is evaluated by the group betweenness centrality. Group betweenness centrality of a group of nodes $C$ is the sum of the fraction of all-pairs shortest paths that pass through any vertex in $C$ .. 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 $\sigma(s, t|C)$ is the number of those paths passing through some node in group $C$. Note that $(s, t)$ are not members of the group ($V-C$ is the set of nodes in $V$ that are not in $C$). Parameters ---------- G : graph A NetworkX graph. k : int The number of nodes in the group. normalized : bool, optional (default=True) If True, group betweenness is normalized by ``1/((|V|-|C|)(|V|-|C|-1))`` where ``|V|`` is the number of nodes in G and ``|C|`` is the number of nodes in C. 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. The weight of an edge is treated as the length or distance between the two sides. endpoints : bool, optional (default=False) If True include the endpoints in the shortest path counts. C : list or set, optional (default=None) list of nodes which won't be candidates of the prominent group. greedy : bool, optional (default=False) Using a naive greedy algorithm in order to find non-optimal prominent group. For scale free networks the results are negligibly below the optimal results. Raises ------ NodeNotFound If node(s) in C are not present in G. Returns ------- max_GBC : float The group betweenness centrality of the prominent group. max_group : list The list of nodes in the prominent group. See Also -------- betweenness_centrality, group_betweenness_centrality Notes ----- Group betweenness centrality is described in [1]_ and its importance discussed in [3]_. The algorithm is described in [2]_ and is based on techniques mentioned in [4]_. The number of nodes in the group must be a maximum of ``n - 2`` where ``n`` is the total number of nodes in the graph. 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 between "u" and "v" are counted as two possible paths (one each direction) while undirected paths between "u" and "v" are counted as one path. Said another way, the sum in the expression above is over all ``s != t`` for directed graphs and for ``s < t`` for undirected graphs. References ---------- .. [1] M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm .. [2] Rami Puzis, Yuval Elovici, and Shlomi Dolev: "Finding the Most Prominent Group in Complex Networks" AI communications 20(4): 287-296, 2007. https://www.researchgate.net/profile/Rami_Puzis2/publication/220308855 .. [3] Sourav Medya et. al.: Group Centrality Maximization via Network Design. SIAM International Conference on Data Mining, SDM 2018, 126–134. https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf .. [4] Rami Puzis, Yuval Elovici, and Shlomi Dolev. "Fast algorithm for successive computation of group betweenness centrality." https://journals.aps.org/pre/pdf/10.1103/PhysRevE.76.056709 """ import numpy as np import pandas as pd if C is not None: C = set(C) if C - G.nodes: # element(s) of C not in G raise nx.NodeNotFound(f"The node(s) {C - G.nodes} are in C but not in G.") nodes = list(G.nodes - C) else: nodes = list(G.nodes) DF_tree = nx.Graph() DF_tree.__networkx_cache__ = None # Disable caching PB, sigma, D = _group_preprocessing(G, nodes, weight) betweenness = pd.DataFrame.from_dict(PB) if C is not None: for node in C: # remove from the betweenness all the nodes not part of the group betweenness.drop(index=node, inplace=True) betweenness.drop(columns=node, inplace=True) CL = [node for _, node in sorted(zip(np.diag(betweenness), nodes), reverse=True)] max_GBC = 0 max_group = [] DF_tree.add_node( 1, CL=CL, betweenness=betweenness, GBC=0, GM=[], sigma=sigma, cont=dict(zip(nodes, np.diag(betweenness))), ) # the algorithm DF_tree.nodes[1]["heu"] = 0 for i in range(k): DF_tree.nodes[1]["heu"] += DF_tree.nodes[1]["cont"][DF_tree.nodes[1]["CL"][i]] max_GBC, DF_tree, max_group = _dfbnb( G, k, DF_tree, max_GBC, 1, D, max_group, nodes, greedy ) v = len(G) if not endpoints: scale = 0 # if the graph is connected then subtract the endpoints from # the count for all the nodes in the graph. else count how many # nodes are connected to the group's nodes and subtract that. if nx.is_directed(G): if nx.is_strongly_connected(G): scale = k * (2 * v - k - 1) elif nx.is_connected(G): scale = k * (2 * v - k - 1) if scale == 0: for group_node1 in max_group: for node in D[group_node1]: if node != group_node1: if node in max_group: scale += 1 else: scale += 2 max_GBC -= scale # normalized if normalized: scale = 1 / ((v - k) * (v - k - 1)) max_GBC *= scale # If undirected then count only the undirected edges elif not G.is_directed(): max_GBC /= 2 max_GBC = float(f"{max_GBC:.2f}") return max_GBC, max_group def _dfbnb(G, k, DF_tree, max_GBC, root, D, max_group, nodes, greedy): # stopping condition - if we found a group of size k and with higher GBC then prune if len(DF_tree.nodes[root]["GM"]) == k and DF_tree.nodes[root]["GBC"] > max_GBC: return DF_tree.nodes[root]["GBC"], DF_tree, DF_tree.nodes[root]["GM"] # stopping condition - if the size of group members equal to k or there are less than # k - |GM| in the candidate list or the heuristic function plus the GBC is below the # maximal GBC found then prune if ( len(DF_tree.nodes[root]["GM"]) == k or len(DF_tree.nodes[root]["CL"]) <= k - len(DF_tree.nodes[root]["GM"]) or DF_tree.nodes[root]["GBC"] + DF_tree.nodes[root]["heu"] <= max_GBC ): return max_GBC, DF_tree, max_group # finding the heuristic of both children node_p, node_m, DF_tree = _heuristic(k, root, DF_tree, D, nodes, greedy) # finding the child with the bigger heuristic + GBC and expand # that node first if greedy then only expand the plus node if greedy: max_GBC, DF_tree, max_group = _dfbnb( G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy ) elif ( DF_tree.nodes[node_p]["GBC"] + DF_tree.nodes[node_p]["heu"] > DF_tree.nodes[node_m]["GBC"] + DF_tree.nodes[node_m]["heu"] ): max_GBC, DF_tree, max_group = _dfbnb( G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy ) max_GBC, DF_tree, max_group = _dfbnb( G, k, DF_tree, max_GBC, node_m, D, max_group, nodes, greedy ) else: max_GBC, DF_tree, max_group = _dfbnb( G, k, DF_tree, max_GBC, node_m, D, max_group, nodes, greedy ) max_GBC, DF_tree, max_group = _dfbnb( G, k, DF_tree, max_GBC, node_p, D, max_group, nodes, greedy ) return max_GBC, DF_tree, max_group def _heuristic(k, root, DF_tree, D, nodes, greedy): import numpy as np # This helper function add two nodes to DF_tree - one left son and the # other right son, finds their heuristic, CL, GBC, and GM node_p = DF_tree.number_of_nodes() + 1 node_m = DF_tree.number_of_nodes() + 2 added_node = DF_tree.nodes[root]["CL"][0] # adding the plus node DF_tree.add_nodes_from([(node_p, deepcopy(DF_tree.nodes[root]))]) DF_tree.nodes[node_p]["GM"].append(added_node) DF_tree.nodes[node_p]["GBC"] += DF_tree.nodes[node_p]["cont"][added_node] root_node = DF_tree.nodes[root] for x in nodes: for y in nodes: dxvy = 0 dxyv = 0 dvxy = 0 if not ( root_node["sigma"][x][y] == 0 or root_node["sigma"][x][added_node] == 0 or root_node["sigma"][added_node][y] == 0 ): if D[x][added_node] == D[x][y] + D[y][added_node]: dxyv = ( root_node["sigma"][x][y] * root_node["sigma"][y][added_node] / root_node["sigma"][x][added_node] ) if D[x][y] == D[x][added_node] + D[added_node][y]: dxvy = ( root_node["sigma"][x][added_node] * root_node["sigma"][added_node][y] / root_node["sigma"][x][y] ) if D[added_node][y] == D[added_node][x] + D[x][y]: dvxy = ( root_node["sigma"][added_node][x] * root_node["sigma"][x][y] / root_node["sigma"][added_node][y] ) DF_tree.nodes[node_p]["sigma"][x][y] = root_node["sigma"][x][y] * (1 - dxvy) DF_tree.nodes[node_p]["betweenness"].loc[y, x] = ( root_node["betweenness"][x][y] - root_node["betweenness"][x][y] * dxvy ) if y != added_node: DF_tree.nodes[node_p]["betweenness"].loc[y, x] -= ( root_node["betweenness"][x][added_node] * dxyv ) if x != added_node: DF_tree.nodes[node_p]["betweenness"].loc[y, x] -= ( root_node["betweenness"][added_node][y] * dvxy ) DF_tree.nodes[node_p]["CL"] = [ node for _, node in sorted( zip(np.diag(DF_tree.nodes[node_p]["betweenness"]), nodes), reverse=True ) if node not in DF_tree.nodes[node_p]["GM"] ] DF_tree.nodes[node_p]["cont"] = dict( zip(nodes, np.diag(DF_tree.nodes[node_p]["betweenness"])) ) DF_tree.nodes[node_p]["heu"] = 0 for i in range(k - len(DF_tree.nodes[node_p]["GM"])): DF_tree.nodes[node_p]["heu"] += DF_tree.nodes[node_p]["cont"][ DF_tree.nodes[node_p]["CL"][i] ] # adding the minus node - don't insert the first node in the CL to GM # Insert minus node only if isn't greedy type algorithm if not greedy: DF_tree.add_nodes_from([(node_m, deepcopy(DF_tree.nodes[root]))]) DF_tree.nodes[node_m]["CL"].pop(0) DF_tree.nodes[node_m]["cont"].pop(added_node) DF_tree.nodes[node_m]["heu"] = 0 for i in range(k - len(DF_tree.nodes[node_m]["GM"])): DF_tree.nodes[node_m]["heu"] += DF_tree.nodes[node_m]["cont"][ DF_tree.nodes[node_m]["CL"][i] ] else: node_m = None return node_p, node_m, DF_tree @nx._dispatchable(edge_attrs="weight") def group_closeness_centrality(G, S, weight=None): r"""Compute the group closeness centrality for a group of nodes. Group closeness centrality of a group of nodes $S$ is a measure of how close the group is to the other nodes in the graph. .. math:: c_{close}(S) = \frac{|V-S|}{\sum_{v \in V-S} d_{S, v}} d_{S, v} = min_{u \in S} (d_{u, v}) where $V$ is the set of nodes, $d_{S, v}$ is the distance of the group $S$ from $v$ defined as above. ($V-S$ is the set of nodes in $V$ that are not in $S$). Parameters ---------- G : graph A NetworkX graph. S : list or set S is a group of nodes which belong to G, for which group closeness centrality is to be calculated. 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. The weight of an edge is treated as the length or distance between the two sides. Raises ------ NodeNotFound If node(s) in S are not present in G. Returns ------- closeness : float Group closeness centrality of the group S. See Also -------- closeness_centrality Notes ----- The measure was introduced in [1]_. The formula implemented here is described in [2]_. Higher values of closeness indicate greater centrality. It is assumed that 1 / 0 is 0 (required in the case of directed graphs, or when a shortest path length is 0). The number of nodes in the group must be a maximum of n - 1 where `n` is the total number of nodes in the graph. For directed graphs, the incoming distance is utilized here. To use the outward distance, act on `G.reverse()`. 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] M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm .. [2] J. Zhao et. al.: Measuring and Maximizing Group Closeness Centrality over Disk Resident Graphs. WWWConference Proceedings, 2014. 689-694. https://doi.org/10.1145/2567948.2579356 """ if G.is_directed(): G = G.reverse() # reverse view closeness = 0 # initialize to 0 V = set(G) # set of nodes in G S = set(S) # set of nodes in group S V_S = V - S # set of nodes in V but not S shortest_path_lengths = nx.multi_source_dijkstra_path_length(G, S, weight=weight) # accumulation for v in V_S: try: closeness += shortest_path_lengths[v] except KeyError: # no path exists closeness += 0 try: closeness = len(V_S) / closeness except ZeroDivisionError: # 1 / 0 assumed as 0 closeness = 0 return closeness @nx._dispatchable def group_degree_centrality(G, S): """Compute the group degree centrality for a group of nodes. Group degree centrality of a group of nodes $S$ is the fraction of non-group members connected to group members. Parameters ---------- G : graph A NetworkX graph. S : list or set S is a group of nodes which belong to G, for which group degree centrality is to be calculated. Raises ------ NetworkXError If node(s) in S are not in G. Returns ------- centrality : float Group degree centrality of the group S. See Also -------- degree_centrality group_in_degree_centrality group_out_degree_centrality Notes ----- The measure was introduced in [1]_. The number of nodes in the group must be a maximum of n - 1 where `n` is the total number of nodes in the graph. References ---------- .. [1] M G Everett and S P Borgatti: The Centrality of Groups and Classes. Journal of Mathematical Sociology. 23(3): 181-201. 1999. http://www.analytictech.com/borgatti/group_centrality.htm """ centrality = len(set().union(*[set(G.neighbors(i)) for i in S]) - set(S)) centrality /= len(G.nodes()) - len(S) return centrality @not_implemented_for("undirected") @nx._dispatchable def group_in_degree_centrality(G, S): """Compute the group in-degree centrality for a group of nodes. Group in-degree centrality of a group of nodes $S$ is the fraction of non-group members connected to group members by incoming edges. Parameters ---------- G : graph A NetworkX graph. S : list or set S is a group of nodes which belong to G, for which group in-degree centrality is to be calculated. Returns ------- centrality : float Group in-degree centrality of the group S. Raises ------ NetworkXNotImplemented If G is undirected. NodeNotFound If node(s) in S are not in G. See Also -------- degree_centrality group_degree_centrality group_out_degree_centrality Notes ----- The number of nodes in the group must be a maximum of n - 1 where `n` is the total number of nodes in the graph. `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph, so for group in-degree centrality, the reverse graph is used. """ return group_degree_centrality(G.reverse(), S) @not_implemented_for("undirected") @nx._dispatchable def group_out_degree_centrality(G, S): """Compute the group out-degree centrality for a group of nodes. Group out-degree centrality of a group of nodes $S$ is the fraction of non-group members connected to group members by outgoing edges. Parameters ---------- G : graph A NetworkX graph. S : list or set S is a group of nodes which belong to G, for which group in-degree centrality is to be calculated. Returns ------- centrality : float Group out-degree centrality of the group S. Raises ------ NetworkXNotImplemented If G is undirected. NodeNotFound If node(s) in S are not in G. See Also -------- degree_centrality group_degree_centrality group_in_degree_centrality Notes ----- The number of nodes in the group must be a maximum of n - 1 where `n` is the total number of nodes in the graph. `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph, so for group out-degree centrality, the graph itself is used. """ return group_degree_centrality(G, S)