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Geometric Routing in Geometric Inhomogeneous Random Graphs

Authors Yu-Cheng Chiu , Marc Kaufmann , Kostas Lakis , Ulysse Schaller

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Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
  • Theory of computation → Graph algorithms analysis
  • Networks → Routing protocols
Keywords
  • geometric inhomogeneous random graphs (GIRGs)
  • hyperbolic random graphs (HRGs)
  • greedy routing
  • geometric routing
  • navigability
  • small-world phenomenon
  • decentralized algorithms

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Abstract

We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length Θ(log log n), it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent τ ∈ (2,3) and geometric decay parameter α > τ-1, geometric routing succeeds with constant probability and finds ultra-short paths of length Θ(log log n), matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network’s geometry alone implicitly guides the path to the target through its high-weight core.

Cite As Get BibTex

Yu-Cheng Chiu, Marc Kaufmann, Kostas Lakis, and Ulysse Schaller. Geometric Routing in Geometric Inhomogeneous Random Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 12:1-12:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.12

Author Details

Yu-Cheng Chiu
  • Department of Mathematics, ETH Zürich, Switzerland
Marc Kaufmann
  • Department of Computer Science, ETH Zürich, Switzerland
Kostas Lakis
  • Department of Computer Science, ETH Zürich, Switzerland
Ulysse Schaller
  • Department of Computer Science, ETH Zürich, Switzerland

Funding

  • Kaufmann, Marc: Swiss National Science Foundation grant number 200021_192079.
  • Lakis, Kostas: Swiss National Science Foundation grant number 10003390.
  • Schaller, Ulysse: Swiss National Science Foundation grant number 200021_192079.

Acknowledgements

We would like to thank Johannes Lengler for suggesting this problem to us and for the insightful conversations about it.

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