Eigenvector centrality

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

${\displaystyle x_{v}={\frac {1}{\lambda }}\sum _{t\in M(v)}x_{t}={\frac {1}{\lambda }}\sum _{t\in G}a_{v,t}x_{t}}$

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

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

In general, there will be many different eigenvalues ${\displaystyle \lambda }$ for which a non-zero eigenvector solution exists. However, the additional requirement that all the entries in the eigenvector be non-negative implies (by the Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure.[5] The ${\displaystyle v^{\text{th}}}$ component of the related eigenvector then gives the relative centrality score of the vertex ${\displaystyle v}$ in the network. The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigen vector e.g. such that the sum over all vertices is 1 or the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector.[4] Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.

Google's PageRank is based on the normalized eigenvector centrality, or normalized prestige, combined with a random jump assumption.[1] The PageRank of a node ${\displaystyle v}$ has recursive dependence on the PageRank of other nodes that point to it. The normalized adjacency matrix ${\displaystyle N}$ is defined as:

$${\displaystyle N(u,v)={\begin{cases}{1 \over \operatorname {od} (u)},&{\text{if }}(u,v)\in E\\0,&{\text{if }}(u,v)\not \in E\end{cases}}}$$

where ${\displaystyle od(u)}$ is the out-degree of node ${\displaystyle u}$. The normalized normalized eigenvector centrality score is defined as:

$${\displaystyle p(v)=\sum _{u}{N^{T}(v,u)\cdot p(u)}}$$

Eigenvector centrality is a measure of the influence a node has on a network. If a node is pointed to by many nodes (which also have high eigenvector centrality) then that node will have high eigenvector centrality.[6]

The earliest use of eigenvector centrality is by Edmund Landau in an 1895 paper on scoring chess tournaments.[7][8]

More recently, researchers across many fields have analyzed applications, manifestations, and extensions of eigenvector centrality in a variety of domains:

• Eigenvector centrality is the unique measure satisfying certain natural axioms for a ranking system.[9][10]
• In neuroscience, the eigenvector centrality of a neuron in a model neural network has been found to correlate with its relative firing rate.[6]
• In a standard class of models of opinion updating or learning (sometimes called DeGroot learning models), the social influence of a node over eventual opinions is equal to its eigenvector centrality.
• The definition of eigenvector centrality has been extended to multiplex or multilayer networks.[11]
• In a economic public goods problems, a person's eigenvector centrality can be interpreted as how much that person's preferences influence an efficient social outcome (formally, a Pareto weight in a Pareto efficient social outcome).[12]

• Centrality