Optimality of Random Regular Graphs in Sparse Network Designs

2026-06-12

AI Summary

This study addresses the fundamental challenge of designing the sparsest network architectures that retain performance comparable to fully connected networks under resource constraints, particularly in the face of uncertain demand or random node failures. By integrating tools from random graph theory, combinatorial optimization, and probabilistic analysis, we establish—for the first time—that random regular graphs achieve a sharp theoretical lower bound on sparsity while preserving near-optimal performance, in both bipartite and non-bipartite settings. Our findings demonstrate that degree regularity combined with low edge correlation is essential for attaining optimal robustness. This work thus identifies random regular graphs as the optimal construction for sparse network design and provides actionable, high-performance design principles for real-world systems operating under uncertainty.

Abstract

The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit -- where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with low edge correlations can achieve optimal performance under uncertainty.

Problem

• sparse networks
• random regular graphs
• network design
• optimality
• uncertainty

Innovation

• random regular graph
• sparse network design
• sharp optimality
• process flexibility
• maximum matching