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Uncertainty quantification and posterior sampling for network reconstruction (2503.07736v1)

Published 10 Mar 2025 in stat.ML, cs.LG, cs.SI, physics.data-an, and physics.soc-ph

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 $O(N\log2 N)$ for a network of $N$ nodes, instead of $O(N2)$, 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.

Summary

An Efficient Algorithm for Network Reconstruction: Uncertainty Quantification and Posterior Sampling

The task of network reconstruction pertains to inferring unseen interactions in complex systems based on observable data. Conventional methods primarily focus on providing point estimates without offering uncertainty quantification. This paper by Tiago P. Peixoto introduces an efficient algorithmic approach using Markov Chain Monte Carlo (MCMC) for sampling posterior distributions with a major focus on representing the plausible network configurations and their associated uncertainties.

Key Contributions

The paper outlines several pivotal contributions in the domain of network reconstruction:

  1. Efficient Posterior Sampling: The approach described involves an efficient MCMC algorithm, capable of sampling from posterior distributions in time proportional to O(Nlog2N)O(N\log^2 N) for a network with NN nodes, as compared to O(N2)O(N^2) in naive methods. This efficiency is particularly beneficial for large and sparse networks.
  2. Uncertainty Quantification: Unlike traditional methods that solely produce point estimates, the proposed methodology provides a full Bayesian posterior sampling, thus allowing for a robust characterization of uncertainties inherent in the reconstructed networks. This aids in understanding the confidence levels associated with different reconstructed network configurations.
  3. Consensus Solutions: The paper emphasizes that the approach leverages consensus over numerous plausible network configurations, weighted by their likelihoods, which invariably enhances reconstruction accuracy compared to point estimates.
  4. Adaptive Quantization: The paper adopts a minimum description length (MDL) principle incorporating adaptive quantization to address the sparsity and complexity in weight distributions, thus optimizing model selection and posterior sampling.

Numerical Results

The exploration of synthetic datasets showcases that the marginal posterior (MP) estimator often outperforms the maximum a posteriori (MAP) point estimate, especially when data is limited or sparse. This is indicative of the MP estimator's ability to optimize the mean squared error effectively, ensuring superior performance in reconstructing the network structure.

Implications and Comparisons

In empirical data settings, the paper compares the probabilistic network reconstructions with heuristic methods based on pairwise correlations, demonstrating that conventional heuristics fall short in accuracy and fail to distinguish between direct and indirect connections adequately. Posterior sampling not only provides enhanced accuracy but also distinguishes between the probability of existence and weight magnitude of edges.

Future Directions

The research paves the way for further investigations into leveraging posterior sampling in realistic generative models beyond simple scenarios discussed in the paper. Investigating the limits of network reconstruction, predictive capabilities, and intervention strategies within network systems through posterior sampling are prospective areas of expansion.

Conclusion

The method introduced in this paper provides significant advancements in network reconstruction, emphasizing efficient posterior sampling and uncertainty quantification, while also demonstrating practical applicability in large-scale empirical data analysis. Researchers in the field should explore the integration of such methodologies in complex systems to improve network inference robustness and accuracy.

Overall, the paper represents notable progress in dealing with the limitations of traditional network reconstruction methods, setting the stage for broader applications and future research endeavors in network science and data analytics.