BUILT-IN MATHEMATICA SYMBOL

EigenvectorCentrality

EigenvectorCentrality[g]
gives a list of eigenvector centralities for the vertices in the graph g.

gives a list of in-centralities for a directed graph g.

EigenvectorCentrality

gives a list of out-centralities for a directed graph g.

MORE INFORMATION

EigenvectorCentrality gives a list of centralities that can be expressed as a weighted sum of centralities of its neighbors.

With being the largest eigenvalue of the adjacency matrix for the graph g you have:

	EigenvectorCentrality[g]
	EigenvectorCentrality[g,"In"]	, left eigenvector
	EigenvectorCentrality[g,"Out"]	, right eigenvector

The option WorkingPrecision->p can be used to control precision used in internal computations.

EXAMPLES

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Basic Examples (2)

Eigenvector centralities for an undirected graph:

Eigenvector centrality for a directed graph:

In-centralities, which are used by default:

Out-centralities:

Eigenvector centralities for an undirected graph:

In[1]:=

Out[1]=

Eigenvector centrality for a directed graph:

In[1]:=

In[2]:=

Out[2]=

In-centralities, which are used by default:

In[3]:=

Out[3]=

Out-centralities:

In[4]:=

Out[4]=

Scope (3)

EigenvectorCentrality for undirected graphs:

Directed graphs:

In-centralities and out-centralities:

Works with large graphs:

Options (3)

By default, EigenvectorCentrality finds centralities using machine-precision computations:

Specify a higher working precision:

Infinite working precision corresponds to exact computation:

Applications (2)

Highlight the eigenvector centrality for CycleGraph:

An unbalanced tree:

Create a random citation graph:

Find the top 10 most central papers:

Highlight the top 10 papers:

Properties & Relations (5)

For undirected graphs, the centrality vector

satisfies the equation

:

Vertices with zero in-degree have centrality zero for directed graphs:

Find the vertices with zero in-degree:

Highlight the zero in-degree vertices and label them by their in-degree:

The eigenvector centralities of all vertices are non-negative:

EigenvectorCentrality is the eigenvector for the largest eigenvalue of its adjacency matrix:

EigenvectorCentrality is a special case of KatzCentrality:

Use

as the parameter for KatzCentrality: