Motif Estimation via Subgraph Sampling: The Fourth Moment Phenomenon

TL;DR

This paper introduces a statistical framework for estimating network motifs from sampled subgraphs, revealing a unique fourth-moment phenomenon that governs the asymptotic normality of estimators across various graph models.

Contribution

It establishes necessary and sufficient conditions for the consistency and asymptotic normality of motif estimators in subgraph sampling, highlighting the fourth-moment phenomenon.

Findings

The HT estimator's asymptotic normality depends on its fourth moment converging to 3.

Derived exact thresholds for estimator properties in different graph models.

Identified a fourth-moment phenomenon influencing estimator distribution.

Abstract

Network sampling is an indispensable tool for understanding features of large complex networks where it is practically impossible to search over the entire graph. In this paper, we develop a framework for statistical inference for counting network motifs, such as edges, triangles, and wedges, in the widely used subgraph sampling model, where each vertex is sampled independently, and the subgraph induced by the sampled vertices is observed. We derive necessary and sufficient conditions for the consistency and the asymptotic normality of the natural Horvitz-Thompson (HT) estimator, which can be used for constructing confidence intervals and hypothesis testing for the motif counts based on the sampled graph. In particular, we show that the asymptotic normality of the HT estimator exhibits an interesting fourth-moment phenomenon, which asserts that the HT estimator (appropriately centered and rescaled) converges in distribution to the standard normal whenever its fourth-moment converges to 3 (the fourth-moment of the standard normal distribution). As a consequence, we derive the exact thresholds for consistency and asymptotic normality of the HT estimator in various natural graph ensembles, such as sparse graphs with bounded degree, Erdos-Renyi random graphs, random regular graphs, and dense graphons.