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+"""Algorithm to select influential nodes in a graph using VoteRank."""
+
+import networkx as nx
+
+__all__ = ["voterank"]
+
+
+@nx._dispatchable
+def voterank(G, number_of_nodes=None):
+ """Select a list of influential nodes in a graph using VoteRank algorithm
+
+ VoteRank [1]_ computes a ranking of the nodes in a graph G based on a
+ voting scheme. With VoteRank, all nodes vote for each of its in-neighbors
+ and the node with the highest votes is elected iteratively. The voting
+ ability of out-neighbors of elected nodes is decreased in subsequent turns.
+
+ Parameters
+ ----------
+ G : graph
+ A NetworkX graph.
+
+ number_of_nodes : integer, optional
+ Number of ranked nodes to extract (default all nodes).
+
+ Returns
+ -------
+ voterank : list
+ Ordered list of computed seeds.
+ Only nodes with positive number of votes are returned.
+
+ Examples
+ --------
+ >>> G = nx.Graph([(0, 1), (0, 2), (0, 3), (1, 4)])
+ >>> nx.voterank(G)
+ [0, 1]
+
+ The algorithm can be used both for undirected and directed graphs.
+ However, the directed version is different in two ways:
+ (i) nodes only vote for their in-neighbors and
+ (ii) only the voting ability of elected node and its out-neighbors are updated:
+
+ >>> G = nx.DiGraph([(0, 1), (2, 1), (2, 3), (3, 4)])
+ >>> nx.voterank(G)
+ [2, 3]
+
+ Notes
+ -----
+ Each edge is treated independently in case of multigraphs.
+
+ References
+ ----------
+ .. [1] Zhang, J.-X. et al. (2016).
+ Identifying a set of influential spreaders in complex networks.
+ Sci. Rep. 6, 27823; doi: 10.1038/srep27823.
+ """
+ influential_nodes = []
+ vote_rank = {}
+ if len(G) == 0:
+ return influential_nodes
+ if number_of_nodes is None or number_of_nodes > len(G):
+ number_of_nodes = len(G)
+ if G.is_directed():
+ # For directed graphs compute average out-degree
+ avgDegree = sum(deg for _, deg in G.out_degree()) / len(G)
+ else:
+ # For undirected graphs compute average degree
+ avgDegree = sum(deg for _, deg in G.degree()) / len(G)
+ # step 1 - initiate all nodes to (0,1) (score, voting ability)
+ for n in G.nodes():
+ vote_rank[n] = [0, 1]
+ # Repeat steps 1b to 4 until num_seeds are elected.
+ for _ in range(number_of_nodes):
+ # step 1b - reset rank
+ for n in G.nodes():
+ vote_rank[n][0] = 0
+ # step 2 - vote
+ for n, nbr in G.edges():
+ # In directed graphs nodes only vote for their in-neighbors
+ vote_rank[n][0] += vote_rank[nbr][1]
+ if not G.is_directed():
+ vote_rank[nbr][0] += vote_rank[n][1]
+ for n in influential_nodes:
+ vote_rank[n][0] = 0
+ # step 3 - select top node
+ n = max(G.nodes, key=lambda x: vote_rank[x][0])
+ if vote_rank[n][0] == 0:
+ return influential_nodes
+ influential_nodes.append(n)
+ # weaken the selected node
+ vote_rank[n] = [0, 0]
+ # step 4 - update voterank properties
+ for _, nbr in G.edges(n):
+ vote_rank[nbr][1] -= 1 / avgDegree
+ vote_rank[nbr][1] = max(vote_rank[nbr][1], 0)
+ return influential_nodes