A systematic association of subgraph counts over a network

TL;DR

This paper introduces a unified graph encoding system that systematically relates small subgraph counts, enabling theoretical insights and practical algorithms for efficient subgraph counting in networks.

Contribution

It develops a comprehensive graph encoding framework connecting graphlet structures with algebraic and numerical relations, and proposes a novel efficient subgraph counting algorithm.

Findings

Established topological relations among graphlets

Characterized algebraic and numerical relations in graphlet counts

Presented an efficient algorithm for small subgraph counting

Abstract

We associate all small subgraph counting problems with a systematic graph encoding/representation system which makes a coherent use of graphlet structures. The system can serve as a unified foundation for studying and connecting many important graph problems in theory and practice. We describe topological relations among graphlets (graph elements) in rigorous mathematics language and from the perspective of graph encoding. We uncover, characterize and utilize algebraic and numerical relations in graphlet counts/frequencies. We present a novel algorithm for efficiently counting small subgraphs as a practical product of our theoretical findings.