Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

TL;DR

The paper investigates how the dimension $d$ of the geometric ground space affects the clique structure of geometric inhomogeneous random graphs (GIRGs). It proves that GIRGs converge to non-geometric inhomogeneous random graphs (IRGs) in total variation as $d\to\infty$ (for $L_p$ norms with $1\le p<\infty$), clarifying when geometry remains detectable. It provides detailed, regime-dependent bounds on the number and size of cliques $K_k$ and the clique number $\omega(G)$ across low and high dimensions, including phase transitions in $k$, $d$, and $\beta$, and shows that high dimensionality eventually yields IRG-like clique behavior, while low dimensionality preserves geometry-driven clustering and heavy-core effects for $\beta\in(2,3)$. The work also establishes concentration results for clique counts and connects the GIRG analysis to prior SRGG/HRG results, resolving some apparent discrepancies related to geometry in different ground spaces. Altogether, the results deepen understanding of how dimensionality governs the interplay between geometry, weights, and clustering in realistic network models and inform when simpler IRG approximations are valid.

Abstract

A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs. In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs.

Figures (1)

• Figure 1: Illustration for the proof of \ref{['thm:prob-high-dim']}. The colored circles represent the sets $A_i$ for $v_1,v_2,v_3$. The probability that $v_2$ is placed in the region indicated by the black circular arc such that $\overline{A}_1 \cap \overline{A}_2 = \emptyset$ is at least $1 - 3(1-2t_0)$ as indicated by the black arrows.

Theorems & Definitions (98)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• Proof 1
• Theorem 2.2: Theorem 2.2 in Keusch_2018, Chernoff-Hoeffding Bound
• Lemma 3.1