Abstract
We derive an explicit formula for the exponential generating function associated with non-backtracking walks around a graph. We study both undirected and directed graphs. Our results allow us to derive computable expressions for non-backtracking versions of network centrality measures based on the matrix exponential. We find that eliminating backtracking walks in this context does not significantly increase the computational expense. We show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network. Insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions. Rigorous analysis on the star graph illustrates the effect of non-backtracking and shows that localization can be eliminated when we restrict to non-backtracking walks. We also investigate the localization issue on synthetic networks.
Highlights
MotivationThe concept of a walk on a graph is very natural—on arriving at a node, the walker may continue by traversing any edge pointing out of that node
We show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network
We add insight by showing how these block matrices may be interpreted directly. We show that they remain relevant when the exponential is replaced by a general matrix function
Summary
The concept of a walk on a graph is very natural—on arriving at a node, the walker may continue by traversing any edge pointing out of that node. Non-backtracking walks have been analyzed in a number of fields They play a key role in the study of zeta functions on graphs [24], with applications in spectral graph theory [4,23], number theory [44], discrete mathematics [12,41], quantum chaos [39], random matrix theory [40], stochastic analysis [3], applied linear algebra [42] and computer science [38,45]. There are effective and reliable tools for computing the action of the matrix exponential [2,21,22] This provides the initial motivation for our article, where we study the exponential generating function associated with non-backtracking walks. We extend the analysis to cover generating functions based on arbitrary power series
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