Ranking hubs and authorities using matrix functions

TL;DR

This paper extends subgraph centrality and communicability measures to directed networks by bipartization, enabling effective ranking of hubs and authorities through matrix exponential techniques and comparison with existing algorithms.

Contribution

It introduces a novel matrix exponential-based method for ranking hubs and authorities in directed networks via bipartization, expanding the applicability of subgraph centrality measures.

Findings

The proposed method effectively ranks hubs and authorities in directed networks.

Comparison shows the matrix exponential method performs well against HITS and other algorithms.

Gaussian quadrature rules facilitate efficient computation of centrality scores.

Abstract

The notions of subgraph centrality and communicability, based on the exponential of the adjacency matrix of the underlying graph, have been effectively used in the analysis of undirected networks. In this paper we propose an extension of these measures to directed networks, and we apply them to the problem of ranking hubs and authorities. The extension is achieved by bipartization, i.e., the directed network is mapped onto a bipartite undirected network with twice as many nodes in order to obtain a network with a symmetric adjacency matrix. We explicitly determine the exponential of this adjacency matrix in terms of the adjacency matrix of the original, directed network, and we give an interpretation of centrality and communicability in this new context, leading to a technique for ranking hubs and authorities. The matrix exponential method for computing hubs and authorities is compared to the well known HITS algorithm, both on small artificial examples and on more realistic real-world networks. A few other ranking algorithms are also discussed and compared with our technique. The use of Gaussian quadrature rules for calculating hub and authority scores is discussed.