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  • Open Access

Fast Nonparametric Inference of Network Backbones for Weighted Graph Sparsification

Alec Kirkley*

  • *Contact author: alec.w.kirkley@gmail.com
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Phys. Rev. X 15, 031013 – Published 11 July, 2025

DOI: https://doi.org/10.1103/4pg6-mtmt

Abstract

Network backbones provide useful sparse representations of weighted networks by keeping only their most important links, permitting a range of computational speedups and simplifying network visualizations. A key limitation of existing network backboning methods is that they either require the specification of a free parameter (e.g., significance level) that determines the number of edges to keep in the backbone or impose specific restrictions on the topology of the backbone (e.g., that it is a spanning tree). Here we develop a completely nonparametric framework for inferring the backbone of a weighted network that overcomes these limitations and automatically selects the optimal set of edges to retain using the minimum description length principle. We develop objective functions for global and local network backboning which evaluate the importance of an edge in the context of the whole network and individual node neighborhoods, respectively, and are generalizable to any weight distribution under Bayesian model specifications that fix the average edge weight either exactly or in expectation. We then construct an efficient and provably optimal greedy algorithm to identify the backbone minimizing our objectives, whose run-time complexity is log-linear in the number of edges. We demonstrate our methods by comparing them with existing methods in a range of tasks on real and synthetic networks, finding that both the global and local backboning methods can preserve network connectivity, weight heterogeneity, and spreading dynamics while removing a substantial fraction of edges.

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Popular Summary

Complex networks—such as social, transportation, or biological networks—often have so many connections that they become difficult to analyze or visualize. To tackle this, researchers use network backboning, a technique that simplifies a network by keeping only the most essential links. A major challenge, however, is deciding how many edges to retain without arbitrarily restricting the network’s structure. In this study, we introduce a new parameter-free method for extracting the backbone of a weighted network using tools from Bayesian inference and information theory.

Our method applies the minimum description length principle, which favors models that offer the shortest, most efficient explanation of the data. Instead of requiring a user to choose a threshold or parameter, our approach automatically finds the backbone that best compresses the network’s information. It does this by striking a balance between the amount of information stored in the backbone and what is left out. The technique is computationally efficient and flexible—it can be used to identify key edges across the whole network and around individual nodes.

This approach not only simplifies network analysis but could also improve our ability to simulate and understand real-world systems. Future work will test how well these backbones preserve network dynamics across various fields such as epidemiology, economics, and transportation.

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Outline

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Phys. Rev. X 15, 031013– Published 11 July, 2025
  • Received 14 October 2024
  • Accepted 13 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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