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A statistical test for network similarity

Pierre Miasnikof, Alexander Y. Shetopaloff

TL;DR

The paper presents a graph comparison framework that avoids node matching by converting graphs into all-pairs Jaccard distance matrices, treating these distances as empirical distributions, and comparing graphs with the Kolmogorov-Smirnov statistic. This probabilistic, multi-scale approach applies to both directed and undirected networks and is demonstrated across synthetic and real-world data, including change-point and anomaly scenarios. The results show that the KS-based distance between distance distributions accurately captures (dis)similarity and remains informative under various structural perturbations, though statistical significance should be interpreted within domain context. The work highlights the method's practicality for monitoring network evolution and its limitations when graphs become disconnected or extremely sparse.

Abstract

In this article, we revisit and expand our prior work on graph similarity. As with our earlier work, we focus on a view of similarity which does not require node correspondence between graphs under comparison. Our work is suited to the temporal study of networks, change-point and anomaly detection and simple comparisons of static graphs. It provides a similarity metric for the study of (weakly) connected graphs. Our work proposes a metric designed to compare networks and assess the (dis)similarity between them. For example, given three different graphs with possibly different numbers of nodes, $G_1$, $G_2$ and $G_3$, we aim to answer two questions: a) "How different is $G_1 $ from $G_2$?" and b) "Is graph $G_3$ more similar to $G_1$ or to $G_2$?". We illustrate the value of our test and its accuracy through several new experiments, using synthetic and real-world graphs.

A statistical test for network similarity

TL;DR

The paper presents a graph comparison framework that avoids node matching by converting graphs into all-pairs Jaccard distance matrices, treating these distances as empirical distributions, and comparing graphs with the Kolmogorov-Smirnov statistic. This probabilistic, multi-scale approach applies to both directed and undirected networks and is demonstrated across synthetic and real-world data, including change-point and anomaly scenarios. The results show that the KS-based distance between distance distributions accurately captures (dis)similarity and remains informative under various structural perturbations, though statistical significance should be interpreted within domain context. The work highlights the method's practicality for monitoring network evolution and its limitations when graphs become disconnected or extremely sparse.

Abstract

In this article, we revisit and expand our prior work on graph similarity. As with our earlier work, we focus on a view of similarity which does not require node correspondence between graphs under comparison. Our work is suited to the temporal study of networks, change-point and anomaly detection and simple comparisons of static graphs. It provides a similarity metric for the study of (weakly) connected graphs. Our work proposes a metric designed to compare networks and assess the (dis)similarity between them. For example, given three different graphs with possibly different numbers of nodes, , and , we aim to answer two questions: a) "How different is from ?" and b) "Is graph more similar to or to ?". We illustrate the value of our test and its accuracy through several new experiments, using synthetic and real-world graphs.

Paper Structure

This paper contains 15 sections, 4 equations, 5 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Previous work
  3. Methods
  4. All-pairs distance
  5. Probabilistic interpretation of the Jaccard distance
  6. From graph to empirical probability distribution
  7. (Dis)similarity of probability distributions
  8. Numerical results
  9. Two-graph tests
  10. Sensitivity analysis
  11. Synthetic graphs
  12. Real-world graphs
  13. Node and vertex additions
  14. Discussion and conclusions
  15. Discussion and conclusions

Figures (5)

  • Figure 1: Distances as distributions. The distribution of node-to-node distances for the Erdős-Rényi (ER) graph is unimodal while it is bimodal for the planted partition model (PPM) graph.
  • Figure 2: Jaccard distance distributions for ER models with given densities.
  • Figure 3: Jaccard distance distributions for stochastic block models with given within block/between block connection probabilities.
  • Figure 4: Jaccard distance distributions for configuration models with 3400 nodes.
  • Figure 5: Pairwise comparisons of CDFs of Jaccard distances.