# EigenvectorCentrality

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

EigenvectorCentrality[g,"In"]

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

EigenvectorCentrality[g,"Out"]

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

EigenvectorCentrality[{vw,},]

uses rules vw to specify the graph g.

# Details and Options • EigenvectorCentrality will give high centralities to vertices that are connected to many other well-connected vertices.
• 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
• Eigenvector centralities are normalized.
• For a directed graph g, is equivalent to EigenvectorCentrality[g,"In"].
• The option can be used to control precision used in internal computations.
• EigenvectorCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Examples

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## Basic Examples(2)

Compute eigenvector centralities:

Highlight:

Rank the vertices. Highest-ranked vertices are connected to many well-connected vertices:

## Scope(7)

EigenvectorCentrality works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Compute in-centralities and out-centralities:

EigenvectorCentrality works with large graphs:

## Options(3)

### WorkingPrecision(3)

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

Specify a higher working precision:

Infinite working precision corresponds to exact computation:

## Applications(9)

Rank vertices by their connectivity to other well-connected vertices:

Order vertices from highest to lowest centrality:

Highlight the eigenvector centrality for CycleGraph:

Find well-connected people in a social network:

Highlight the network:

Find the most influential members of a student government network with connections from student A to student B, if student A consults student B for opinions:

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

Find hubs of the internet at the level of autonomous systems:

Hubs are well-connected vertices and have the highest eigenvector centralities:

Find a protein whose deletion will result in lethality in a protein interaction network of yeast:

Protein with the largest eigenvector centrality:

Deleting the protein increases the diameter of the network:

A Saccharomyces cerevisiae protein interaction network. The frequency of the eigenvector centrality follows a power-law distribution:

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

Probability density function:

A human-computer system of an organization that deals with internet orders and sends goods by mail. Find departments that should be given the most resources:

Most services rely on a system administration department as well as a register orders department:

## Properties & Relations(6)

For undirected graphs, the centrality vector satisfies the equation :

For directed graphs, in-centrality vector satisfies the equation :

Out-centrality vector satisfies the equation :

Eigenvector centralities are normalized:

For disconnected graphs, centralities are normalized with respect to the connected components:

Centrality of the subgraph generated by the connected component:

EigenvectorCentrality is a special case of KatzCentrality:

Use and as the parameters for KatzCentrality:

Use VertexIndex to obtain the centrality of a specific vertex: