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On Connectivity in Random Graph Models with Limited Dependencies

Authors Johannes Lengler , Anders Martinsson , Kalina Petrova , Patrick Schnider , Raphael Steiner , Simon Weber , Emo Welzl

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Johannes Lengler
  • Department of Computer Science, ETH Zürich, Switzerland
Anders Martinsson
  • Department of Computer Science, ETH Zürich, Switzerland
Kalina Petrova
  • Department of Computer Science, ETH Zürich, Switzerland
Patrick Schnider
  • Department of Computer Science, ETH Zürich, Switzerland
Raphael Steiner
  • Department of Computer Science, ETH Zürich, Switzerland
Simon Weber
  • Department of Computer Science, ETH Zürich, Switzerland
Emo Welzl
  • Department of Computer Science, ETH Zürich, Switzerland


This research started at the joint workshop of the Combinatorial Structures and Algorithms and Theory of Combinatorial Algorithms groups of ETH Zürich held in Stels, Switzerland, January 2023. We thank the organizers for providing a very pleasant and inspiring working atmosphere. We thank Victor Falgas-Ravry for pointing out an abundance of related work, thus helping tremendously to put our results in perspective.

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Johannes Lengler, Anders Martinsson, Kalina Petrova, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl. On Connectivity in Random Graph Models with Limited Dependencies. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 30:1-30:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


For any positive edge density p, a random graph in the Erdős-Rényi G_{n,p} model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability ρ(n), such that for any distribution 𝒢 (in this model) on graphs with n vertices in which each potential edge has a marginal probability of being present at least ρ(n), a graph drawn from 𝒢 is connected with non-zero probability? As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold ρ(n). For each condition, we provide upper and lower bounds for ρ(n). In the strongest condition, the coloring model (which includes, e.g., random geometric graphs), we show that ρ(n) → 2-ϕ ≈ 0.38 for n → ∞, proving a conjecture by Badakhshian, Falgas-Ravry, and Sharifzadeh. This separates the coloring models from the weaker independence conditions we consider, as there we prove that ρ(n) > 0.5-o(n). In stark contrast to the coloring model, for our weakest independence condition - pairwise independence of non-adjacent edges - we show that ρ(n) lies within O(1/n²) of the threshold 1-2/n for completely arbitrary distributions.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Theory of computation → Randomness, geometry and discrete structures
  • Random Graphs
  • Independence
  • Dependency
  • Connectivity
  • Threshold
  • Probabilistic Method


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