A Theory for Backtrack-Downweighted Walks

TL;DR

This paper develops a comprehensive mathematical framework for analyzing walk-counting on directed graphs with backtracking steps downweighted, enabling efficient computation of centrality measures that generalize existing models and improve network analysis.

Contribution

It introduces a complete theory for backtrack-downweighted walks, deriving generating functions, linear systems for centrality, and new eigenvector centrality expressions, unifying and extending prior models.

Findings

Backtrack-downweighted Katz centrality can be computed as efficiently as standard Katz.

The new eigenvector centrality generalizes previous models to directed graphs.

The theory effectively combines advantages of standard and nonbacktracking centralities.

Abstract

We develop a complete 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 localization while accounting for tree-like structures. We illustrate the behaviour of the backtrack-downweighted centrality measure on both synthetic and real networks.