Network properties revealed through matrix functions

Ernesto Estrada, Desmond Higham

Research output: Contribution to journalArticle

158 Citations (Scopus)

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.
Original languageEnglish
Pages (from-to)696-714
Number of pages19
JournalSIAM Review
Volume52
Issue number4
DOIs
Publication statusPublished - 8 Nov 2010

Keywords

  • centrality measures
  • clustering methods
  • communicability
  • Estrada index
  • Fiedler vector
  • graph Laplacian
  • graph spectrum
  • power series
  • resolvent

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