Measuring Similarity of Graphs and their Nodes by Neighbor Matching

The problem of measuring similarity of graphs and their nodes is important in a range of practical problems. There is a number of proposed measures, some of them being based on iterative calculation of similarity between two graphs and the principle that two nodes are as similar as their neighbors are. In our work, we propose one novel method of that sort, with a refined concept of similarity of two nodes that involves matching of their neighbors. We prove convergence of the proposed method and show that it has some additional desirable properties that, to our knowledge, the existing methods lack. We illustrate the method on two specific problems and empirically compare it to other methods.

💡 Research Summary

The paper addresses the fundamental problem of quantifying similarity between whole graphs and between individual nodes within graphs, a task that underlies many applications ranging from cheminformatics to social network analysis. While a variety of similarity measures have been proposed, most of them follow the “neighbor‑similarity” principle: two nodes are deemed similar if their neighbors are similar. Existing approaches such as SimRank, graph kernels, and Weisfeiler‑Lehman based methods either treat neighbor sets as unordered bags, ignore the size disparity between neighbor sets, or lack rigorous convergence guarantees. Moreover, many of them assume symmetry, which is inappropriate for directed or weighted graphs where the relationship between two nodes can be inherently asymmetric.

The authors propose a novel iterative similarity measure that explicitly matches the neighbors of two nodes. For a pair of nodes (u) and (v) with neighbor sets (N(u)) and (N(v)), the similarity at iteration (t+1) is defined as

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