The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs

TL;DR

This paper establishes that the diameter of threshold Geometric Inhomogeneous Random Graphs (T-GIRG) is $\Theta(\log n)$, with implications for the runtimes of distance- and diameter-related distributed protocols. The authors develop a high-dimensional renormalization approach based on a hierarchical tessellation of the ground space into boxes, paired with canonical box paths and the boundary-connectivity framework to control connectivity. They show that, whp, any two connected vertices can be joined by a path largely contained in a logarithmically bounded region $W(u,v) \cup S(u,v)$, and bound the size of this region via a Peierls-type argument. In particular, for the regime $\tau\in(2,3)$ (and $\tau=3$ with large $\lambda$), the diameter scales as $\Theta(\log n)$, aligning with properties observed in real networks and supporting GIRG as a proxy for understanding algorithmic performance on large networks. The work introduces novel higher-dimensional techniques beyond prior 1D approaches used for HRGs and paves the way for analyzing more general threshold random geometric models.

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.

Figures (1)

• Figure 1: Visualization of the tessellation for $d = 2$, with $3$ levels. The side length of lowest-level boxes is $D_{0}$. The blue boxes comprise $\mathcal{T}_{0}$ (weight range $[1, 2)$), the green ones are $\mathcal{T}_{1}$ (weight range $[2, 4)$) and the red box is $\mathcal{T}_{2}$ (weight range $[4, \infty)$). Also depicted is a vertex path arising from a fully active canonical box path (as in \ref{['def:canonboxpath']}).

Theorems & Definitions (5)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Definition 2.1
• Definition 2.2: Canonical box path