Estimating Graphlet Statistics via Lifting

TL;DR

This paper presents Lifting, a novel Monte Carlo sampling framework for estimating small subgraph counts (graphlets) in large networks efficiently, with provable unbiasedness and variance control.

Contribution

Introduces Lifting, the first graphlet sampling method that can uniformly sample all graphlets of arbitrary size with positive probability and efficiency.

Findings

Lifting provides unbiased estimates of graphlet counts.

Lifting outperforms existing methods like Waddling and pairwise subgraph random walk.

The method is applicable to massive graphs with limited neighborhood queries.

Abstract

Exploratory analysis over network data is often limited by the ability to efficiently calculate graph statistics, which can provide a model-free understanding of the macroscopic properties of a network. We introduce a framework for estimating the graphlet count---the number of occurrences of a small subgraph motif (e.g. a wedge or a triangle) in the network. For massive graphs, where accessing the whole graph is not possible, the only viable algorithms are those that make a limited number of vertex neighborhood queries. We introduce a Monte Carlo sampling technique for graphlet counts, called {\em Lifting}, which can simultaneously sample all graphlets of size up to $k$ vertices for arbitrary $k$. This is the first graphlet sampling method that can provably sample every graphlet with positive probability and can sample graphlets of arbitrary size $k$. We outline variants of lifted graphlet counts, including the ordered, unordered, and shotgun estimators, random walk starts, and parallel vertex starts. We prove that our graphlet count updates are unbiased for the true graphlet count and have a controlled variance for all graphlets. We compare the experimental performance of lifted graphlet counts to the state-of-the art graphlet sampling procedures: Waddling and the pairwise subgraph random walk.