Faster motif counting via succinct color coding and adaptive sampling

TL;DR

This paper introduces optimized color coding and adaptive sampling techniques for faster, more accurate motif counting in large graphs, enabling the detection of rare motifs efficiently.

Contribution

It extends color coding applicability with algorithmic optimizations and a novel adaptive sampling scheme that improves accuracy and scalability for large graph motif counting.

Findings

Built a compact table for 8-node motifs in one hour on large graphs

Reduced running time and space usage significantly compared to previous methods

Counted rare motifs with high accuracy, impossible with uniform sampling

Abstract

We address the problem of computing the distribution of induced connected subgraphs, aka \emph{graphlets} or \emph{motifs}, in large graphs. The current state-of-the-art algorithms estimate the motif counts via uniform sampling, by leveraging the color coding technique by Alon, Yuster and Zwick. In this work we extend the applicability of this approach, by introducing a set of algorithmic optimizations and techniques that reduce the running time and space usage of color coding and improve the accuracy of the counts. To this end, we first show how to optimize color coding to efficiently build a compact table of a representative subsample of all graphlets in the input graph. For $8$-node motifs, we can build such a table in one hour for a graph with $65$M nodes and $1.8$B edges, which is $2000$ times larger than the state of the art. We then introduce a novel adaptive sampling scheme that breaks the "additive error barrier" of uniform sampling, guaranteeing multiplicative approximations instead of just additive ones. This allows us to count not only the most frequent motifs, but also extremely rare ones. For instance, on one graph we accurately count nearly $10.000$ distinct $8$-node motifs whose relative frequency is so small that uniform sampling would literally take centuries to find them. Our results show that color coding is still the most promising approach to scalable motif counting.