Graph Measures & Metrics

Mathematica supports a broad range of measures that characterize graphs, from simple measures, such as the number of vertices and edges that tell the size and sparsity of a graph, to vertex degrees, which tell how locally well-connected each vertex is. Other measures include the geodesic distances in a graph or centrality measures that give a measure of how central in the overall graph each vertex is; for example, PageRank and HITS are measures used to order web page importance as returned from a search engine.


Basic Measures

VertexCount the number of vertices

EdgeCount the number of edges

Degree Measures

VertexDegree the number of edges for each vertex

VertexInDegree the number of in-edges for each vertex

VertexOutDegree the number of out-edges for each vertex

Distance Measures

GraphDistance the length of the shortest path between two vertices

GraphDistanceMatrix ▪ VertexEccentricity ▪ GraphRadius ▪ GraphDiameter

Connectivity Measures

VertexConnectivity the number of vertex independent paths between two vertices

EdgeConnectivity the number of edge independent paths between two vertices

Centrality Measures

ClosenessCentrality inverse average distance to every other vertex

BetweennessCentrality fraction of shortest paths that pass through the vertex

DegreeCentrality ▪ EigenvectorCentrality ▪ KatzCentrality ▪ PageRankCentrality ▪ HITSCentrality ▪ RadialityCentrality ▪ StatusCentrality ▪ EdgeBetweennessCentrality

Reciprocity and Transitivity

GraphReciprocity fraction of directed edges that are reciprocated

GlobalClusteringCoefficient fraction of length two paths that are closed

MeanClusteringCoefficient ▪ LocalClusteringCoefficient

Homophily, Assortative Mixing, and Similarity

GraphAssortativity within-groups connectivity minus between-group connectivity

VertexCorrelationSimilarity correlation similarity between actors

MeanNeighborDegree ▪ MeanDegreeConnectivity ▪ VertexDiceSimilarity ▪ VertexJaccardSimilarity ▪ VertexCosineSimilarity

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