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|≥...≥|λN| = 0 correspond to the normalized eigenvectors X1 , X2 , ... ,XN. 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 = [λ12 ... λN]' be the vector of eigenvalues. Furthermore, if P < N and if matrix X has dimensions N x N , then XNxP 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:
Principal Component Centrality
Above equation can also be written in terms of the eigenvalue and eigenvector matrices Λ and X, of the adjacency matrix A:
Principal Component Centrality

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.


Good day,

Wonderful work, I have been searching all over the internet on principal component centrality computation and here it is in your website. kindly assist me on how to implement this particalar centrality using either the centiserver or centiserve please, my email address is I look forward to hearing from you soon. Thank you.

Murtala Sheriff.

Add Replay written April 27, 2016, 3:00 am by Sheriff Murtala

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