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Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs

Authors Tobias Friedrich , Andreas Göbel , Maximilian Katzmann, Leon Schiller

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Author Details

Tobias Friedrich
  • Hasso Plattner Institute, Universität Potsdam, Germany
Andreas Göbel
  • Hasso Plattner Institute, Universität Potsdam, Germany
Maximilian Katzmann
  • Karlsruhe Institute of Technology, Germany
Leon Schiller
  • Hasso Plattner Institute, Universität Potsdam, Germany

Funding

  • Göbel, Andreas: is funded by the project PAGES (project No. 467516565) of the German Research Foundation (DFG).

Cite AsGet BibTex

Tobias Friedrich, Andreas Göbel, Maximilian Katzmann, and Leon Schiller. Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.62

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Theory of computation → Computational geometry
Keywords
  • random graphs
  • geometry
  • dimensionality
  • cliques
  • clique number
  • scale-free networks

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