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arXiv · 2511.14058

On the number of small edge-weighted subgraphs

Feng Yu·Mingao Yuan
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Abstract

Subgraph counting is a fundamental task that underpins several network analysis methodologies, including community detection and graph two-sample tests. Counting subgraphs is a computationally intensive problem. Substantial research has focused on developing efficient algorithms and strategies to make it feasible for larger unweighted graphs. Implementing those algorithms can be a significant hurdle for data professionals or researchers with limited expertise in algorithmic principles and programming. Furthermore, many real-world networks are weighted. Computing the number of weighted subgraphs in weighted networks presents a computational challenge, as no efficient algorithm exists for the worst-case scenario. In this paper, we derive explicit formulas for counting small edge-weighted subgraphs using the weighted adjacency matrix. These formulas are applicable to unweighted networks, offering a simple and highly practical analytical tool for researchers across various scientific domains. In addition, we introduce a generalized methodology for calculating arbitrary weighted subgraphs.

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Feng Yu, Mingao Yuan. 2026-08-07. On the number of small edge-weighted subgraphs. https://arxiv.org/abs/2511.14058

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