A Consistent Histogram Estimator for Exchangeable Graph Models

TL;DR

This paper introduces a scalable, provably consistent histogram estimator for graphons in exchangeable graph models, utilizing a sorting-and-smoothing approach that combines degree sorting with total variation minimization.

Contribution

It presents a novel, efficient estimator for graphons that is mathematically proven to be consistent, addressing a key challenge in network modeling.

Findings

The estimator is provably consistent.

The method is numerically efficient.

It leverages compressed sensing concepts.

Abstract

Exchangeable graph models (ExGM) subsume a number of popular network models. The mathematical object that characterizes an ExGM is termed a graphon. Finding scalable estimators of graphons, provably consistent, remains an open issue. In this paper, we propose a histogram estimator of a graphon that is provably consistent and numerically efficient. The proposed estimator is based on a sorting-and-smoothing (SAS) algorithm, which first sorts the empirical degree of a graph, then smooths the sorted graph using total variation minimization. The consistency of the SAS algorithm is proved by leveraging sparsity concepts from compressed sensing.