A theory for backtrack-downweighted walks

F. Arrigo, D. J. Higham, V. Noferini

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We develop a comprehensive theory for the combinatorics of walk-counting on a directed graph in the case where each backtracking step is downweighted by a given factor. By deriving expressions for the associated generating functions, we also obtain linear systems for computing centrality measures in this setting. In particular, we show that backtrack-downweighted Katz-style network centrality can be computed at the same cost as standard Katz. Studying the limit of this centrality measure at its radius of convergence also leads to a new expression for backtrack-downweighted eigenvector centrality that generalizes previous work to the case where directed edges are present. The new theory allows us to combine advantages of standard and nonbacktracking cases, avoiding certain types of localization while accounting for tree-like structures. We illustrate the behaviour of the backtrack-downweighted centrality measure on both synthetic and real networks.
Original languageEnglish
Number of pages19
JournalSIAM Journal on Matrix Analysis and Applications
Issue number3
Publication statusPublished - 30 Sept 2021


  • centrality index
  • complex network
  • localization
  • nonbacktracking walk
  • generating function
  • zeta function


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