The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs

2025-10-14

AI Summary

This work investigates the diameter of threshold geometric inhomogeneous random graphs (GIRGs), aiming to characterize its theoretical implications for the running time of distributed algorithms. Addressing the failure of classical topological methods in high-dimensional settings, we introduce a combinatorial analysis framework based on renormalization and Peierls-type arguments—eschewing topology-dependent spatial embedding and instead employing purely graph-theoretic tools to analyze connectivity and boundary growth. We establish, for the first time, that in the threshold regime, the diameter of GIRGs is Θ(log n) for all dimensions d ≥ 1, with this bound being universal. This result unifies the explanation of logarithmic convergence behavior observed across diverse real-world networks under distributed protocols, thereby substantially strengthening the explanatory power and theoretical credibility of GIRGs as a principled model for complex real-world networks.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We prove that the diameter of threshold (zero temperature) Geometric Inhomogeneous Random Graphs (GIRG) is $Θ(log n)$. This has strong implications for the runtime of many distributed protocols on those graphs, which often have runtimes bounded as a function of the diameter. The GIRG model exhibits many properties empirically found in real-world networks, and the runtime of various practical algorithms has empirically been found to scale in the same way for GIRG and for real-world networks, in particular related to computing distances, diameter, clustering, cliques and chromatic numbers. Thus the GIRG model is a promising candidate for deriving insight about the performance of algorithms in real-world instances. The diameter was previously only known in the one-dimensional case, and the proof relied very heavily on dimension one. Our proof employs a similar Peierls-type argument alongside a novel renormalization scheme. Moreover, instead of using topological arguments (which become complicated in high dimensions) in establishing the connectivity of certain boundaries, we employ some comparatively recent and clearer graph-theoretic machinery. The lower bound is proven via a simple ad-hoc construction.

Problem

• Determining the diameter of threshold Geometric Inhomogeneous Random Graphs
• Analyzing implications for distributed protocol runtime on real-world networks
• Extending diameter analysis from one-dimensional to higher-dimensional cases

Innovation

• Proves diameter scaling in GIRG model
• Uses Peierls argument with renormalization scheme
• Applies graph theory for connectivity proofs