On the number of small edge-weighted subgraphs

TL;DR

This paper introduces explicit formulas and a generalized methodology for counting small edge-weighted subgraphs in weighted networks, addressing a key computational challenge in network analysis.

Contribution

It provides the first explicit formulas for small weighted subgraph counting and a new general approach for arbitrary weighted subgraphs, simplifying analysis in weighted networks.

Findings

Explicit formulas for small weighted subgraph counts

A generalized method for arbitrary weighted subgraphs

Applicable to unweighted networks as a special case

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.