Principal Component Centrality


Definition

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.

Software

References

  • 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.


Comments

Good day,

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Murtala Sheriff.

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