[Paper Review] A Bayesian approach to out-of-sample network reconstruction

Mattia Marzi, Squartini, Tiziano|arXiv (Cornell University)|Feb 25, 2026

Complex Systems and Time Series Analysis0 citations

TL;DR

The paper develops a Bayesian framework to reconstruct and predict evolving networks using past snapshots as priors, enabling out-of-sample inference with quantified uncertainty, demonstrated on the eMID interbank market.

ABSTRACT

Networks underpin systems that range from finance to biology, yet their structure is often only partially observed. Current reconstruction methods typically fit the parameters of a model anew to each snapshot, thus offering no guidance to predict future configurations. Here, we develop a Bayesian approach that uses the information about past network snapshots to inform a prior and predict the subsequent ones, while quantifying uncertainty. Instantiated with a single-parameter fitness model, our method infers link probabilities from node strengths and carries information forward in time. When applied to the Electronic Market for Interbank Deposit across the years 1999-2012, our method accurately recovers the number of connections per bank at subsequent times, outperforming probabilistic benchmarks designed for analogous, link prediction tasks. Notably, each predicted snapshot serves as a reliable prior for the next one, thus enabling self-sustained, out-of-sample reconstruction of evolving networks with a minimal amount of additional data.

Motivation & Objective

• Motivate the reconstruction of partially observed networks and the need for out-of-sample predictions.
• Propose a Bayesian extension of the Undirected Binary Configuration Model to propagate information over time.
• Introduce a single-parameter Bayesian models (BERM and BFM) to infer future link probabilities from past data.
• Demonstrate the approach on the eMID interbank market to predict future topology and assess uncertainty.

Proposed method

• Transform the Undirected Binary Configuration Model into a Bayesian framework using P(x|A) and the posterior predictive distribution P(A_{t+1}|A_t).
• Derive analytical expressions for edge-conditional probabilities q_{ij}^{t+1} under marginalization over z, enabling forward inference (Eq. 9).
• Instantiate the Bayesian Erdős–Rényi Model (BERM) with Beta priors to obtain a Beta-Binomial predictive for L_{t+1}.
• Instantiate the Bayesian Fitness Model (BFM) based on the density-corrected Gravity Model with node strengths as fitnesses and a prior on z derived from empirical history.

Experimental results

Research questions

• RQ1 How can past network snapshots inform priors for predicting future network configurations?
• RQ2 Can a Bayesian formulation enable reliable out-of-sample network reconstruction with uncertainty quantification?
• RQ3 Do single-parameter, heterogeneous models (BERM and BFM) outperform in-sample methods in predicting future links and preserving degree sequences?
• RQ4 How does self-sustained inference perform when the model uses only its own past predictions as priors?
• RQ5 What is the comparative predictive performance of Bayesian approaches versus in-sample dcGM on real financial network data?

Key findings

• Both BERM and BFM recover the total number of links over time; the BFM better captures degree heterogeneity.
• BFM provides non-trivial link ranking enabling ROC/AUROC and Jaccard index based evaluation, unlike BERM which treats pairs uniformly.
• BFM achieves higher accuracy and better degree sequence recovery than BERM, with ACC around 0.80 on average but higher ranking metrics for BFM.
• Self-sustained inference, where priors evolve from predicted ensembles rather than observed networks, yields predictions close to those using true adjacency matrices.
• Compared to in-sample dcGM, the Bayesian predictor matches performance on average and outperforms it on a substantial share of snapshots, especially for edge-level metrics.