ESCAPE: Efficiently Counting All 5-Vertex Subgraphs

TL;DR

This paper introduces ESCAPE, a scalable algorithmic framework for exact counting of all 5-vertex subgraphs in large graphs, significantly advancing the efficiency and practicality of small pattern counting in network analysis.

Contribution

The paper presents a novel framework that reduces 5-vertex pattern counting to smaller subgraph counts, enabling practical analysis of massive graphs with unprecedented speed.

Findings

Counts all 5-vertex subgraphs in graphs with tens of millions of edges in minutes

The framework reduces the problem to counting only four specific subgraphs

The 4-vertex counting method is ten times faster than previous state-of-the-art algorithms

Abstract

Counting the frequency of small subgraphs is a fundamental technique in network analysis across various domains, most notably in bioinformatics and social networks. The special case of triangle counting has received much attention. Getting results for 4-vertex or 5-vertex patterns is highly challenging, and there are few practical results known that can scale to massive sizes. We introduce an algorithmic framework that can be adopted to count any small pattern in a graph and apply this framework to compute exact counts for \emph{all} 5-vertex subgraphs. Our framework is built on cutting a pattern into smaller ones, and using counts of smaller patterns to get larger counts. Furthermore, we exploit degree orientations of the graph to reduce runtimes even further. These methods avoid the combinatorial explosion that typical subgraph counting algorithms face. We prove that it suffices to enumerate only four specific subgraphs (three of them have less than 5 vertices) to exactly count all 5-vertex patterns. We perform extensive empirical experiments on a variety of real-world graphs. We are able to compute counts of graphs with tens of millions of edges in minutes on a commodity machine. To the best of our knowledge, this is the first practical algorithm for $5$-vertex pattern counting that runs at this scale. A stepping stone to our main algorithm is a fast method for counting all $4$-vertex patterns. This algorithm is typically ten times faster than the state of the art $4$-vertex counters.