Wavelet Latent Position Exponential Random Graphs

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.

💡 Research Summary

The paper introduces Wavelet Latent Position Exponential Random Graphs (WL‑ERGs), a novel exchangeable graphon model designed for networks where each vertex possesses an observable coordinate (e.g., geographic location, genomic position, anatomical coordinates). The core idea is to represent the log‑odds of the edge‑probability kernel in an orthonormal wavelet basis on the unit interval, then map these log‑odds to probabilities via a logistic link. Wavelet coefficients are indexed by both resolution (scale) and location, which yields a sparse, interpretable multiscale parameterization.

Formally, a compactly supported orthonormal wavelet family {ψ_r} (e.g., Haar or Daubechies) is fixed on (0,1). For each vertex i a latent position U_i∼Uniform(0,1) is drawn, and the feature vector φ(U_i) = (ψ_0(U_i), ψ_1(U_i), …)^T is constructed. A symmetric Hilbert‑Schmidt matrix S = (s_{rs}) defines the bilinear form f_S(x,y)=φ(x)^T S φ(y). The log‑odds η_{c,S}(x,y)=c+f_S(x,y) is passed through σ(t)=1/(1+e^{−t}) to obtain the edge probability W_{c,S}(x,y). Conditional on the latent positions, edges are independent Bernoulli(W_{c,S}(U_i,U_j)).

When S is truncated to a finite index set I, the conditional distribution of the adjacency matrix belongs to a finite‑dimensional exponential family with sufficient statistics S_{rs}(A,U)=∑{i<j} A{ij} ψ_r(U_i) ψ_s(U_j). This yields a maximum‑entropy characterization and enables regularization (e.g., L1, L2 penalties) and hypothesis testing directly in coefficient space.

The authors prove several key theoretical results:

1. Universality – Any logistic graphon can be expressed via an appropriate wavelet‑logit parameterization; special cases recover Erdős–Rényi, stochastic block models (SBM) when only coarse Haar scaling coefficients are non‑zero, and logistic random dot product graphs (RDPG) when S is low‑rank positive semidefinite.
2. Near‑minimax estimation – Under a multiscale sparsity assumption (only a small number of coefficients are non‑zero), a penalized likelihood estimator attains the minimax rate up to logarithmic factors. The authors derive explicit detection thresholds: a coefficient of magnitude larger than √(log n / n) can be consistently detected at its corresponding scale.
3. Band‑limited regime – When the expansion is truncated to a fixed resolution, the resulting finite‑dimensional exponential family is non‑degenerate, and a finite‑dimensional large‑deviation principle holds, providing precise asymptotics for likelihood‑based tests.
4. Parameter‑complexity separations – The paper quantifies how many wavelet coefficients are needed to approximate a given graphon within ε error, showing that WL‑ERGs can achieve the same approximation error with far fewer parameters than pure block or low‑rank models when the true structure exhibits localized fine‑scale deviations.

Algorithmically, the latent positions are either estimated (e.g., via order statistics) or treated as known in a “observed‑design” setting. The coefficient matrix S is estimated by maximizing the conditional likelihood with sparsity‑inducing penalties, using an ADMM scheme for scalability. Model selection across scales is performed via information criteria (AIC/BIC) combined with multiple‑testing corrections.

Empirical evaluations include synthetic benchmarks where WL‑ERGs recover planted multiscale patterns more accurately than SBM or low‑rank RDPG baselines, and two real‑world case studies: (i) structural brain connectomes, where high‑resolution wavelet coefficients pinpoint subtle fronto‑posterior connectivity variations missed by coarse models; (ii) village social networks in Karnataka, India, where mid‑resolution coefficients align with geographic clusters, demonstrating the model’s ability to capture both global and local spatial effects. In both cases, WL‑ERGs achieve higher predictive AUC and provide interpretable visualizations of where and at what scale connectivity deviates from baseline.

In summary, WL‑ERGs fuse wavelet multiresolution analysis with exchangeable graphon theory, delivering a statistically principled, computationally tractable framework that simultaneously accommodates coarse block structure, smooth global geometry, and localized fine‑scale anomalies. The work opens avenues for extensions to sparse graphs, dynamic multilayer networks, and fully Bayesian treatments with wavelet‑sparse priors.