Accounting for the Role of Long Walks on Networks via a New Matrix Function

TL;DR

This paper introduces a novel matrix function based on double-factorial penalization to analyze walks in networks, revealing differences in network properties, especially in networks with chordless cycles, and providing new tools for network analysis.

Contribution

The paper proposes a new matrix function based on the matrix error function and hyperbolic tangent approximation, enabling improved analysis of long walks in networks with structural features like chordless cycles.

Findings

The new matrix function effectively captures long walks in networks.

Significant differences in node rankings are observed when using double-factorial penalization.

Networks with chordless cycles show distinct properties under the new penalization scheme.

Abstract

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.