Statistical network isomorphism

TL;DR

This paper introduces a statistical method for comparing network structures by transforming graphs into distance matrices, modeling these as probability distributions, and applying statistical tests to assess similarity or difference, applicable across various fields.

Contribution

It proposes a novel statistical framework for network comparison based on node-node distance distributions, moving beyond simple graph metrics.

Findings

The method accurately detects network similarities and differences.

Validation on synthetic and real-world graphs supports its effectiveness.

It outperforms traditional graph comparison metrics.

Abstract

Graph isomorphism is a problem for which there is no known polynomial-time solution. Nevertheless, assessing (dis)similarity between two or more networks is a key task in many areas, such as image recognition, biology, chemistry, computer and social networks. Moreover, questions of similarity are typically more general and their answers more widely applicable than the more restrictive isomorphism question. In this article, we offer a statistical answer to the following questions: a) {\it ``Are networks $G_1$ and $G_2$ similar?''}, b) {\it ``How different are the networks $G_1$ and $G_2$?''} and c) {\it ``Is $G_3$ more similar to $G_1$ or $G_2$?''}. Our comparisons begin with the transformation of each graph into an all-pairs distance matrix. Our node-node distance, Jaccard distance, has been shown to offer a good reflection of the graph's connectivity structure. We then model these distances as probability distributions. Finally, we use well-established statistical tools to gauge the (dis)similarities in terms of probability distribution (dis)similarity. This comparison procedure aims to detect (dis)similarities in connectivity structure, not in easily observable graph characteristics, such as degrees, edge counts or density. We validate our hypothesis that graphs can be meaningfully summarized and compared via their node-node distance distributions, using several synthetic and real-world graphs. Empirical results demonstrate its validity and the accuracy of our comparison technique.