Centrality Measures
Centrality measures describe the importance of a vertex to the rest of the graph using some set of criteria. Centrality measures implemented in Erdos.jl include the following:
Erdos.betweenness_centralityErdos.closeness_centralityErdos.coresErdos.degree_centralityErdos.in_degree_centralityErdos.katz_centralityErdos.kcoreErdos.out_degree_centralityErdos.pagerank
Erdos.betweenness_centrality — Methodbetweenness_centrality(g; normalize=true, endpoints=false, approx=-1)Calculates the betweenness centrality of the vertices of graph g.
Betweenness centrality for vertex v is defined as:
\[bc(v) = \frac{1}{\mathcal{N}} \sum_{s \neq t \neq v} \frac{\sigma_{st}(v)}{\sigma_{st}},\]
where $\sigma _{st}} \sigma_{st}$ is the total number of shortest paths from node s to node t and $\sigma_{st}(v)$ is the number of those paths that pass through v.
If endpoints=true, endpoints are included in the shortest path count.
If normalize=true, the betweenness values are normalized by the total number of possible distinct paths between all pairs in the graph. For an undirected graph, this number if ((n-1)*(n-2))/2 and for a directed graph, (n-1)*(n-2) where n is the number of vertices in the graph.
If an integer argument approx > 0 is given, returns an approximation of the betweenness centrality of each vertex of the graph involving approx randomly chosen vertices.
References
[1] Brandes 2001 & Brandes 2008
Erdos.closeness_centrality — MethodCalculates the closeness centrality of the graph g.
Erdos.degree_centrality — MethodCalculates the degree centrality of the graph g, with optional (default) normalization.
Erdos.in_degree_centrality — MethodCalculates the degree centrality of the graph g, with optional (default) normalization.
Erdos.out_degree_centrality — MethodCalculates the degree centrality of the graph g, with optional (default) normalization.
Erdos.katz_centrality — FunctionCalculates the Katz centrality of the graph g.
Erdos.pagerank — Functionpagerank(g::ADiGraph, α=0.85, n=100, ϵ = 1.0e-6)Calculates the PageRank of the graph g. Can optionally specify a different damping factor (α), number of iterations (n), and convergence threshold (ϵ). If convergence is not reached within n iterations, an error will be returned.
Erdos.cores — Methodcores(g)Returns a vector deg such that if deg[v]=k then the vertex v belongs to the k-core of g and not to the k+1-core.
See also kcore.
Erdos.kcore — Methodkcore(g, k) -> (gnew, vmap)Returns the k-core of g along with a vertex map associating the mutated vertex indexes to the old ones (as in rem_vertices!).
See also cores