Accounting for the role of long walks on networks via a new matrix function

Ernesto Estrada, Grant Silver

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)
41 Downloads (Pure)


We introduce a new matrix function for studying graphs and real-world networks based on a double-factorial penalization of walks between nodes in a graph. This new matrix function is based on the matrix error function. We find a very good approximation of this function using a matrix hyperbolic tangent function. We derive a communicability function, a subgraph centrality and a double-factorial Estrada index based on this new matrix function. We obtain upper and lower bounds for the double-factorial Estrada index of graphs, showing that they are similar to those of the single-factorial Estrada index. We then compare these indices with the single-factorial one for simple graphs and real-world networks. We conclude that for networks containing chordless cycles-holes-the two penalization schemes produce significantly different results. In particular, we study two series of real-world networks representing urban street networks, and protein residue networks. We observe that the subgraph centrality based on both indices produce significantly different ranking of the nodes. The use of the double factorial penalization of walks opens new possibilities for studying important structural properties of real-world networks where long-walks play a fundamental role, such as the cases of networks containing chordless cycles.
Original languageEnglish
Pages (from-to)1581-1600
Number of pages20
JournalJournal of Mathematical Analysis and Applications
Issue number2
Early online date3 Jan 2017
Publication statusPublished - 15 May 2017


  • matrix error functions
  • matrix tanh function
  • communicability functions
  • double-factorial
  • chordless cycles
  • complex networks


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