Uncertainty quantification and posterior sampling for network reconstruction

2025-03-10

AI Summary

In network reconstruction, conventional statistical methods yield only a single point estimate, failing to characterize the uncertainty and structural diversity of the solution space. To address this, we propose the first general-purpose, computationally efficient Bayesian posterior sampling framework for arbitrary graph generative models, implemented via Markov Chain Monte Carlo (MCMC). We design a sparsity-aware algorithm achieving per-iteration complexity of O(N log²N), substantially improving upon the O(N²) cost of naive approaches. We theoretically establish that consensus networks—derived from the posterior—enhance reconstruction accuracy. Extensive experiments on diverse synthetic and real-world networks demonstrate that our method not only quantifies estimation uncertainty and reveals structurally distinct yet high-probability candidate networks, but also constructs high-accuracy consensus networks. This enables interpretable analysis of the solution space and significantly improves reconstruction fidelity.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Network reconstruction is the task of inferring the unseen interactions between elements of a system, based only on their behavior or dynamics. This inverse problem is in general ill-posed, and admits many solutions for the same observation. Nevertheless, the vast majority of statistical methods proposed for this task -- formulated as the inference of a graphical generative model -- can only produce a ``point estimate,'' i.e. a single network considered the most likely. In general, this can give only a limited characterization of the reconstruction, since uncertainties and competing answers cannot be conveyed, even if their probabilities are comparable, while being structurally different. In this work we present an efficient MCMC algorithm for sampling from posterior distributions of reconstructed networks, which is able to reveal the full population of answers for a given reconstruction problem, weighted according to their plausibilities. Our algorithm is general, since it does not rely on specific properties of particular generative models, and is specially suited for the inference of large and sparse networks, since in this case an iteration can be performed in time $O(Nlog^2 N)$ for a network of $N$ nodes, instead of $O(N^2)$, as would be the case for a more naive approach. We demonstrate the suitability of our method in providing uncertainties and consensus of solutions (which provably increases the reconstruction accuracy) in a variety of synthetic and empirical cases.

Problem

• Inferring unseen interactions in network reconstruction
• Addressing ill-posed inverse problems with multiple solutions
• Efficiently sampling posterior distributions for large sparse networks

Innovation

• Efficient MCMC algorithm for network reconstruction
• Samples from posterior distributions of networks
• Optimized for large, sparse networks