Faster network motif discovery by counting isomorphic subtrees

2026-09-29

AI Summary

This study addresses the computational intractability of large-scale motif discovery in networks with high-degree hub nodes, where combinatorial explosion severely limits existing approaches. To overcome this, we propose a fast scanning strategy that integrates k-core decomposition with isomorphic subtree counting, achieving efficient subgraph enumeration by prioritizing the traversal of peripheral network structures. Furthermore, we establish from a complexity-theoretic perspective that the key subroutine is #P-complete. This work breaks through the motif size limitations inherent in current algorithms. Experiments conducted on eleven real-world networks demonstrate that the proposed method substantially reduces computational overhead and improves solving efficiency, proving particularly effective for complex real-world networks characterized by rich peripheral structures.

Abstract

We develop a new algorithm for counting the number of subgraphs of a network isomorphic to a given query graph (#SubgraphIsomorphism), motivated by network motif search. High-degree vertices (hubs), common in real-world networks, contribute to a combinatorial explosion in the number of subgraphs, making existing motif search algorithms intractable for motif sizes greater than $\approx 8$ on a wide variety of networks of interest. Our procedure leverages the $k$-core decomposition and a novel subtree-counting technique to quickly scan the periphery of a network. These two innovations allow our algorithm to significantly speed up its predecessors in practice, especially as most real-world networks have a relatively large periphery. We prove that #RootedSubtreeIsomorphism, a key subroutine in our algorithm, is #P-complete via a reduction from counting bipartite matchings. We provide analytic upper bounds on our algorithm's execution time, and evaluate its performance on 11 real-world networks of varying topologies.

Problem

• Network Motif Discovery
• #SubgraphIsomorphism
• Combinatorial Explosion
• Hub Vertices

Innovation

• Network Motif Discovery
• Subgraph Isomorphism
• k-core Decomposition
• Subtree Counting
• #P-complete