MATHEMATICA GUIDE

# 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.

## ReferenceReference

### 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

### 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

### Reciprocity and Transitivity

GraphReciprocity fraction of directed edges that are reciprocated

GlobalClusteringCoefficient fraction of length two paths that are closed

### Homophily, Assortative Mixing, and Similarity

GraphAssortativity within-groups connectivity minus between-group connectivity

VertexCorrelationSimilarity correlation similarity between actors

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