Principal Component Centrality

Principal Component Centrality (PCC) of a node in a graph as the Euclidean distance/l² norm of a node from the origin in the P-dimensional eigenspace formed by the P most significant eigenvectors. For a graph consisting of a single connected component, the N eigenvalues |λ₁|≥|λ₂|≥...≥|λ_N| = 0 correspond to the normalized eigenvectors X₁ , X₂ , ... ,X_N. The eigenvector/eigenvalue pairs are indexed in order of descending magnitude of eigenvalues.

Let X denote the N x N matrix of concatenated eigenvectors X = [X1X2 ... XN] and let Λ = [λ₁,λ₂ ... λ_N]' be the vector of eigenvalues. Furthermore, if P < N and if matrix X has dimensions N x N , then X_NxP will denote the submatrix of X consisting of the first N rows and first P columns. Then PCC can be expressed in matrix form as: Above equation can also be written in terms of the eigenvalue and eigenvector matrices Λ and X, of the adjacency matrix A:
PCC is a measure of node centrality and is based on PCA and the Karhunen Loeve transform (KLT) which takes the view of treating a graphs adjacency matrix as a covariance matrix.
Unlike eigenvector centrality, PCC allows the addition of more features for the computation of node centralities.

Weighted Principal Component Centrality

Li, J.R., YU, L. and ZHAO, J., 2014. A Node Centrality Evaluation Model for Weighted Social Networks. Journal of University of Electronic Science and Technology of China, 43(3), pp.322-328.

• ILYAS, M. U. & RADHA, H. A KLT-inspired node centrality for identifying influential neighborhoods in graphs. Information Sciences and Systems (CISS), 2010 44th Annual Conference on, 17-19 March 2010 2010. 1-7.