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The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs

Authors Zylan Benjert, Kostas Lakis , Johannes Lengler , Raghu Raman Ravi

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LIPIcs.STACS.2026.11.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Distributed algorithms
Keywords
  • GIRG
  • Diameter
  • Distributed Algorithms
  • Complex Networks

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Abstract

We prove that the diameter of threshold (zero temperature) Geometric Inhomogeneous Random Graphs (GIRG) is asymptotically almost surely 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.

Cite As Get BibTex

Zylan Benjert, Kostas Lakis, Johannes Lengler, and Raghu Raman Ravi. The Diameter of (Threshold) Geometric Inhomogeneous Random Graphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 11:1-11:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.11

Author Details

Zylan Benjert
  • TU Delft, Netherlands
Kostas Lakis
  • ETH Zurich, Switzerland
Johannes Lengler
  • ETH Zurich, Switzerland
Raghu Raman Ravi
  • ETH Zurich, Switzerland

Funding

K.L. and R.R.R. gratefully acknowledge support by the Swiss National Science Foundation [grant number 10003390].

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