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Network properties revealed through matrix functions

Estrada, Ernesto and Higham, Desmond (2010) Network properties revealed through matrix functions. SIAM Review, 52 (4). pp. 696-714.

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Abstract

The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, communicability, and betweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network's adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks.

Item type: Article
ID code: 29071
Keywords: centrality measures, clustering methods, communicability, Estrada index, Fiedler vector, graph Laplacian, graph spectrum, power series, resolvent, Probabilities. Mathematical statistics, Computational Mathematics, Theoretical Computer Science, Applied Mathematics
Subjects: Science > Mathematics > Probabilities. Mathematical statistics
Department: Faculty of Science > Mathematics and Statistics
Related URLs:
Depositing user: Pure Administrator
Date Deposited: 17 Mar 2011 14:28
Last modified: 27 Mar 2014 09:14
URI: http://strathprints.strath.ac.uk/id/eprint/29071

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