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


gives a list of vertex in-degrees.


gives a list of vertex out-degrees.


uses rules vw to specify the graph g.


  • 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.


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Basic Examples  (2)

Compute degree centralities:


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

Scope  (7)

DegreeCentrality works with undirected graphs:

Directed graphs:


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:

Wolfram Research (2010), DegreeCentrality, Wolfram Language function, (updated 2015).


Wolfram Research (2010), DegreeCentrality, Wolfram Language function, (updated 2015).


Wolfram Language. 2010. "DegreeCentrality." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015.


Wolfram Language. (2010). DegreeCentrality. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_degreecentrality, author="Wolfram Research", title="{DegreeCentrality}", year="2015", howpublished="\url{}", note=[Accessed: 14-June-2024 ]}


@online{reference.wolfram_2024_degreecentrality, organization={Wolfram Research}, title={DegreeCentrality}, year={2015}, url={}, note=[Accessed: 14-June-2024 ]}