# Implement Eigenvector Centrality

Registered by Dimitris Kalamaras on 2010-01-10

In eigenvector centrality, we must calculate the principal eigenvector of the network. In this respect, a node is central to the extent that its neighbors are central.

Using the adjacency matrix to find eigenvector centrality

For a given graph G:=(V,E) with |V| number of vertices let A = (a_{v,t}) be the adjacency matrix, i.e. a_{v,t} = 1 if vertex v is linked to vertex t, and a_{v,t} = 0 otherwise. The centrality score of vertex v can be defined as:

x_v = \frac{1}{\lambda} \sum_{t \in M(v)}x_t = \frac{1}{\lambda} \sum_{t \in G} a_{v,t}x_t

where M(v) is a set of the neighbors of v and \lambda is a constant. With a small rearrangement this can be rewritten in vector notation as the eigenvector equation

\mathbf{Ax} = {\lambda}\mathbf{x}

In general, there will be many different eigenvalues \lambda for which an eigenvector solution exists. However, the additional requirement that all the entries in the eigenvector be positive implies (by the Perronâ€“Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure.[19] The v^{th} component of the related eigenvector then gives the centrality score of the vertex v in the network. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector.[18] Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.

## Blueprint information

Status:
Complete
Approver:
Priority:
Essential
Drafter:
Direction:
Approved
Assignee:
Definition:
Approved
Series goal:
Accepted for 2.x
Implementation:
Implemented
Milestone target:
2.2
Started by
Dimitris Kalamaras on 2017-01-23
Completed by
Dimitris Kalamaras on 2017-07-16

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