# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# DegreeCentrality

gives a list of vertex degrees for the vertices in the underlying simple graph of g.

DegreeCentrality[g,"In"]
gives a list of vertex in-degrees.

DegreeCentrality[g,"Out"]
gives a list of vertex out-degrees.

DegreeCentrality[{vw,},]
uses rules to specify the graph g.

## DetailsDetails

• DegreeCentrality will give high centralities to vertices that have high vertex degrees.
• The vertex degree for a vertex is the number of edges incident to .
• For a directed graph, the in-degree is the number of incoming edges and the out-degree is the number of outgoing edges.
• For an undirected graph, in-degree and out-degree coincide.
• DegreeCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

## Background & ContextBackground & Context

• DegreeCentrality returns a list of non-negative integers ("degree centralities") that are particular centrality measures of the vertices of a graph. Degree centrality is a measure of the centrality of a node in a network and is defined as the number of edges (including self-loops) that lead into or out of the node. Degree centralities therefore lie between and inclusive, where is the number of vertices in a graph, and identify nodes in the network by their influence on other nodes in their immediate neighborhood. This measure has found applications in social networks, transportation, biology, and the social sciences.
• A second argument "In" or "Out" can be used to give a list of vertex in- or out-degrees, respectively, which correspond to one another for undirected graphs.
• DegreeCentrality is a local measure that is equivalent to VertexDegree for simple graphs. Similar centrality measures include EigenvectorCentrality and KatzCentrality.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Compute degree centralities:

 Out[2]=

Highlight:

 Out[3]=

Rank vertices. Highest-ranked vertices have the most connections to other vertices:

 Out[2]=