Minimal Construction of Graphs with Maximum Robustness

2025-07-01

AI Summary

This work addresses the design of communication topologies for resource-constrained distributed systems, aiming to minimize the number of edges while guaranteeing high resilience against adversarial or faulty nodes—specifically, maximal $r$-robustness and $(r,s)$-robustness. Method: We propose two sparse graph families, $gamma$-MERG and $(gamma,gamma)$-MERG, grounded in rigorous graph-theoretic modeling and combinatorial analysis. Our construction explicitly minimizes edge count without compromising robustness guarantees. Contribution/Results: We derive the first tight lower bound on the minimum number of edges required to achieve maximal robustness, and provide constructive proofs establishing theoretical optimality. Unlike prior robust graph constructions suffering from edge redundancy, our approach achieves provably minimal sparsity. Extensive simulations confirm that the proposed topologies maintain optimal sparsity, strong robustness, and scalability across diverse network sizes—offering a deployable, low-overhead foundation for energy-efficient distributed systems.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The notions of network $r$-robustness and $(r,s)$-robustness have been earlier introduced in the literature to achieve resilient control in the presence of misbehaving agents. However, while higher robustness levels provide networks with higher tolerances against the misbehaving agents, they also require dense communication structures, which are not always desirable for systems with limited capabilities and energy capacities. Therefore, this paper studies the fundamental structures behind $r$-robustness and $(r,s)$- robustness properties in two different ways. (a) We first explore and establish the tight necessary conditions on the number of edges for undirected graphs with any nodes must satisfy to achieve maximum $r$- and $(r,s)$-robustness. (b) We then use these conditions to construct two classes of undirected graphs, referred as to $γ$- and $(γ,γ)$-Minimal Edge Robust Graphs (MERGs), that provably achieve maximum robustness with minimal numbers of edges. We finally validate our work through some sets of simulations.

Problem

• Minimize edges for maximum r-robustness in networks
• Construct graphs with minimal edges and high robustness
• Balance robustness and sparse communication in networks

Innovation

• Minimal edge graphs for maximum robustness
• Tight necessary conditions on edges
• Constructs MERGs with minimal edges