How can we measure how well important nodes are connected to other important nodes? For example, one might have one friend in the world, but this friend is the president of the USA. This vertex is important; it has an important friend.
The eigenvector centrality of a vertex \(i\) is the sum of the eigenvector centrality of its neighbors \(j\): \[ x_i = \lambda^{-1} \sum_{j \in N(i)} x_j \] where \(\lambda\) is the constant of proportionality.
Solving this equation for all nodes is equivalent to solving the eigenvalue problem for the largest eigenvalue, where the matrix is the network's adjacency matrix.
Let \(G\) be a connected undirected graph with adjacency matrix \(\mathbf{A}\), for convenience defined as: \[ A_{ij} = \begin{cases} 1 & \text{ if there is a node } j \text{ pointing to node } i\\ 0 & \text{ otherwise} \end{cases} \] Eigenvector Centrality: Solve the eigenvalue equation \[ \mathbf{x} = \lambda^{-1} \mathbf{A} \mathbf{x} \] The \(ith\) component of the eigenvector with the largest eigenvalue is the eigenvector centrality of the node \(i\).