Embedding networks with the random walk first return time distribution

TL;DR

This paper introduces the first return time distribution (FRTD) as a novel, interpretable node embedding method for complex networks, which captures structural similarity and outperforms traditional metrics in network alignment tasks.

Contribution

The paper presents FRTD as a new mathematically grounded node embedding that encodes structural information and improves network analysis beyond existing spectral methods.

Findings

FRTD is more informative than eigenvalue spectra.

FRTD-based embedding outperforms manual graph metrics in network alignment.

Matching FRTD in random networks preserves key features.

Abstract

We propose the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded node embedding. The FRTD assigns a probability mass function to each node, allowing us to define a distance between any pair of nodes using standard metrics for discrete distributions. We present several arguments to motivate the FRTD embedding. First, we show that FRTDs are strictly more informative than eigenvalue spectra, yet insufficient for complete graph identification, thus placing FRTD equivalence between cospectrality and isomorphism. Second, we argue that FRTD equivalence between nodes captures structural similarity. Third, we empirically demonstrate that the FRTD embedding outperforms manually designed graph metrics in network alignment tasks. Finally, we show that random networks that approximately match the FRTD of a desired target also preserve other salient features. Together these results demonstrate the FRTD as a simple and mathematically principled embedding for complex networks.