# 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 vw to specify the graph g.

# Details • 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.

# Examples

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## Basic Examples(2)

Compute degree centralities:

Highlight:

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

## Scope(7)

DegreeCentrality works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Compute in-degree and out-degree:

DegreeCentrality works with large graphs:

## Applications(8)

Rank vertices by their influence on other vertices in their immediate neighborhood:

Highlight the degree centrality for CycleGraph:

A friendship network in a school. Find the most popular students:

A citation network from the High Energy Physics Phenomenology section of the arXiv e-Print archive. Find the top 10 most-cited articles:

Find the basal species or producers in a food chain:

Find the top species or apex predators:

A network of email sent to the MathGroup list in November 2011. Construct a social network of users, with an edge from to if has sent at least one reply to . Find the users who are the most active at answering questions:

Count the users who only received replies and did not send any replies:

Find the users who asked questions most often:

Count the users who only sent replies and did not receive any replies:

The internet at the level of autonomous systems. The frequency of the degree centrality follows a power-law distribution:

Obtain the maximum likelihood parameter estimates, assuming a Zipf distribution:

Probability density function:

For graphs with vertices, the largest sum in differences in degree centrality between the most central vertex and all other vertices is :

Measure how central the most central vertex is with respect to other vertices:

Centralization of social networks:

## Properties & Relations(5)

The degree of a vertex of an undirected graph is the number of edges incident to the vertex:

For an undirected graph, the in-degree and out-degree centralities coincide:

Use VertexDegree to obtain the degree of a specific vertex:

DegreeCentrality is equivalent to VertexDegree for simple graphs:

For a directed graph, the sum of in- and out-degree centralities is equal to the vertex degree: