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# EigenvectorCentrality

 EigenvectorCentrality[g] gives a list of eigenvector centralities for the vertices in the graph g. EigenvectorCentralitygives a list of in-centralities for a directed graph g. EigenvectorCentralitygives a list of out-centralities for a directed graph g.
• 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.
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:
 Out[1]=

Eigenvector centrality for a directed graph:
 Out[2]=
In-centralities, which are used by default:
 Out[3]=
Out-centralities:
 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:
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:
Use as the parameter for KatzCentrality:
New in 8