Wavelet Latent Position Exponential Random Graphs

TL;DR

This paper introduces Wavelet Latent Position ERGs, a multiscale network model using wavelet transforms to capture and interpret complex connectivity patterns across different resolutions.

Contribution

It develops a novel wavelet-based framework for exponential random graphs that enables multiscale, interpretable, and regularized modeling of network connectivity.

Findings

Wavelet coefficients provide sparse, interpretable multiscale network features.

The model admits a maximum-entropy characterization with conditional laws.

The framework supports likelihood-based regularization and hypothesis testing.

Abstract

Many network datasets exhibit connectivity with variance by resolution and large-scale organization that coexists with localized departures. When vertices have observed ordering or embedding, such as geography in spatial and village networks, or anatomical coordinates in connectomes, learning where and at what resolution connectivity departs from a baseline is crucial. Standard models typically emphasize a single representation, i.e. stochastic block models prioritize coarse partitions, latent space models prioritize global geometry, small-world generators capture local clustering with random shortcuts, and graphon formulations are fully general and do not solely supply a canonical multiresolution parameterization for interpretation and regularization. We introduce wavelet latent position exponential random graphs (WL-ERGs), an exchangeable logistic-graphon framework in which the log-odds connectivity kernel is represented in compactly supported orthonormal wavelet coordinates and mapped to edge probabilities through a logistic link. Wavelet coefficients are indexed by resolution and location, which allows multiscale structure to become sparse and directly interpretable. Although edges remain independent given latent coordinates, any finite truncation yields a conditional exponential family whose sufficient statistics are multiscale wavelet interaction counts and conditional laws admit a maximum-entropy characterization. These characteristics enable likelihood-based regularization and testing directly in coefficient space. The theory is naturally scale-resolved and includes universality for broad classes of logistic graphons, near-minimax estimation under multiscale sparsity, scale-indexed recovery and detection thresholds, and a band-limited regime in which canonical coefficient-space tilts are non-degenerate and satisfy a finite-dimensional large deviation principle.