Walk Entropies in Graphs

TL;DR

This paper introduces walk entropies based on graph and line-graph properties, revealing their relation to graph regularity, and demonstrates their effectiveness in correlating with eigenmode participation ratios.

Contribution

It defines new walk entropies derived from the thermal Green's function, highlighting their advantages over existing measures and exploring their temperature dependence.

Findings

Walk entropies correlate better with eigenmode participation ratios.

Walk entropies are non-monotonic in regular but non-walk-regular graphs.

They are not biased by graph size and relate to walk regularity.

Abstract

Entropies based on walks on graphs and on their line-graphs are defined. They are based on the summation over diagonal and off-diagonal elements of the thermal Green's function of a graph also known as the communicability. The walk entropies are strongly related to the walk regularity of graphs and line-graphs. They are not biased by the graph size and have significantly better correlation with the inverse participation ratio of the eigenmodes of the adjacency matrix than other graph entropies. The temperature dependence of the walk entropies is also discussed. In particular, the walk entropy of graphs is shown to be non-monotonic for regular but non-walk-regular graphs in contrast to non-regular graphs.