Wavelet Latent Position Exponential Random Graphs

2025-12-20

AI Summary

Existing models struggle to simultaneously characterize multiscale connectivity baselines and localized anomalous deviations in ordered node networks (e.g., geographic or neuroanatomical networks). Method: We propose the first sparse logistic-graphon framework based on compactly supported orthogonal wavelet bases, enabling interpretable, regularizable, and testable scale-resolved modeling. It employs a logistic link function and conditional exponential-family structure to construct multiscale sufficient statistics, supporting maximum-entropy inference in coefficient space and large-deviation analysis. Contributions/Results: We establish theoretical universality—covering a broad class of logistic graph models—and prove near-minimax estimation accuracy under multiscale sparsity. Furthermore, we derive scale-dependent thresholds for structural recovery and anomaly detection, providing rigorous statistical guarantees for both inference tasks. The framework unifies interpretability, regularization, and hypothesis testing within a wavelet-based multiscale paradigm.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

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.

Problem

• Model multiscale network connectivity with interpretable wavelet coefficients.
• Detect localized departures from baseline connectivity across resolutions.
• Enable likelihood-based regularization and testing in coefficient space.

Innovation

• Wavelet coefficients index resolution and location for sparsity
• Logistic link maps wavelet kernel to edge probabilities
• Multiscale wavelet interaction counts enable likelihood-based regularization