The Wolfram Language 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-group connectivity minus between-group connectivity

VertexCorrelationSimilarity — correlation similarity between actors

MeanNeighborDegree  ▪  MeanDegreeConnectivity  ▪  VertexDiceSimilarity  ▪  VertexJaccardSimilarity  ▪  VertexCosineSimilarity