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# On Connectivity in Random Graph Models with Limited Dependencies

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## Acknowledgements

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.

## Cite As

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)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.30

## Abstract

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
##### Keywords
• Random Graphs
• Independence
• Dependency
• Connectivity
• Threshold
• Probabilistic Method

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## References

1. Jon Aaronson, David Gilat, Michael Keane, and Vincent de Valk. An algebraic construction of a class of one-dependent processes. The annals of probability, pages 128-143, 1989.
2. Noga Alon and Asaf Nussboim. k-wise independent random graphs. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 813-822, 2008. URL: https://doi.org/10.1109/FOCS.2008.61.
3. Noga Alon and Joel H. Spencer. The probabilistic method. John Wiley & Sons, 2016. URL: https://doi.org/10.1002/9780470277331.
4. Leila Badakhshian, Victor Falgas-Ravry, and Maryam Sharifzadeh. On density conditions for transversal trees in multipartite graphs. arXiv preprint, 2023. URL: https://doi.org/10.48550/arXiv.2305.05713.
5. Paul Balister, Béla Bollobás, and Mark Walters. Random transceiver networks. Advances in Applied Probability, 41(2):323-343, 2009. URL: https://doi.org/10.1239/aap/1246886613.
6. Paul Balister, Tom Johnston, Michael Savery, and Alex Scott. Improved bounds for 1-independent percolation on ℤⁿ. arXiv preprint, 2022. URL: https://doi.org/10.48550/arXiv.2206.12335.
7. Béla Bollobás. The isoperimetric number of random regular graphs. European Journal of Combinatorics, 9(3):241-244, 1988. URL: https://doi.org/10.1016/S0195-6698(88)80014-3.
8. Adrian Bondy, Jian Shen, Stéphan Thomassé, and Carsten Thomassen. Density conditions for triangles in multipartite graphs. Combinatorica, 26(2):121-131, 2006. URL: https://doi.org/10.1007/s00493-006-0009-y.
9. Karl Bringmann, Ralph Keusch, and Johannes Lengler. Geometric inhomogeneous random graphs. Theoretical Computer Science, 760:35-54, 2019. URL: https://doi.org/10.1016/j.tcs.2018.08.014.
10. Robert M. Burton, Marc Goulet, and Ronald Meester. On 1-dependent processes and k-block factors. The Annals of Probability, 21(4):2157-2168, 1993. URL: https://doi.org/10.1214/aop/1176989014.
11. Constantin Carathéodory. Über den variabilitätsbereich der koeffizienten von potenzreihen, die gegebene werte nicht annehmen. Mathematische Annalen, 64(1):95-115, 1907. URL: https://doi.org/10.1007/BF01449883.
12. Fan Chung and Linyuan Lu. Connected components in random graphs with given expected degree sequences. Annals of combinatorics, 6(2):125-145, 2002. URL: https://doi.org/10.1007/PL00012580.
13. Péter Csikvári and Zoltán L. Nagy. The density Turán problem. Combinatorics, Probability and Computing, 21(4):531-553, 2012. URL: https://doi.org/10.1017/S0963548312000016.
14. A. Nicholas Day, Victor Falgas-Ravry, and Robert Hancock. Long paths and connectivity in 1-independent random graphs. Random Structures & Algorithms, 57(4):1007-1049, 2020. URL: https://doi.org/10.1002/rsa.20972.
15. Maria Deijfen, Remco van der Hofstad, and Gerard Hooghiemstra. Scale-free percolation. Annales de l'IHP Probabilités et statistiques, 49(3):817-838, 2013. URL: https://doi.org/10.1214/12-AIHP480.
16. Paul Erdős. Graph theory and probability. Canadian Journal of Mathematics, 11:34-38, 1959. URL: https://doi.org/10.4153/CJM-1959-003-9.
17. Victor Falgas-Ravry and Vincent Pfenninger. 1-independent percolation on ℤ² × K_n. Random Structures & Algorithms, 2022. URL: https://doi.org/10.1002/rsa.21129.
18. Heidi Gebauer, Robin A. Moser, Dominik Scheder, and Emo Welzl. The Lovász Local Lemma and satisfiability. In Susanne Albers, Helmut Alt, and Stefan Näher, editors, Efficient Algorithms: Essays Dedicated to Kurt Mehlhorn on the Occasion of His 60th Birthday, pages 30-54. Springer, Berlin, Heidelberg, 2009. URL: https://doi.org/10.1007/978-3-642-03456-5_3.
19. Alexander E. Holroyd and Thomas M. Liggett. Finitely dependent coloring. In Forum of Mathematics, Pi, volume 4, page e9. Cambridge University Press, 2016. URL: https://doi.org/10.1017/fmp.2016.7.
20. Anders Johansson. Asymptotic choice number for triangle free graphs. Technical Report, DIMACS, pages 91-95, 1996.
21. Brian Karrer and Mark E. J. Newman. Stochastic blockmodels and community structure in networks. Physical review E, 83(1):016107, 2011. URL: https://doi.org/10.1103/PhysRevE.83.016107.
22. Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010. URL: https://doi.org/10.1103/PhysRevE.82.036106.
23. Thomas M. Liggett, Roberto H. Schonmann, and Alan M. Stacey. Domination by product measures. The Annals of Probability, 25(1):71-95, 1997. URL: https://doi.org/10.1214/aop/1024404279.
24. Michael Molloy. The list chromatic number of graphs with small clique number. Journal of Combinatorial Theory, Series B, 134(1):264-284, 2019. URL: https://doi.org/10.1016/j.jctb.2018.06.007.
25. Zoltán L. Nagy. A multipartite version of the Turán problem - density conditions and eigenvalues. The Electronic Journal of Combinatorics, 18(1):P46, 2011. URL: https://doi.org/10.37236/533.
26. Mathew Penrose. Random Geometric Graphs. Oxford University Press, 2003. URL: https://doi.org/10.1093/acprof:oso/9780198506263.001.0001.
27. Hannu Reittu and Ilkka Norros. Random graph models of communication network topologies. In Unifying Themes in Complex Systems VII: Proceedings of the Seventh International Conference on Complex Systems, pages 214-221. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-18003-3_21.
28. James B. Shearer. On a problem of Spencer. Combinatorica, 5(3):241-245, 1985. URL: https://doi.org/10.1007/BF02579368.
29. Joel Spencer. Asymptotic lower bounds for Ramsey functions. Discrete Mathematics, 20:69-76, 1977. URL: https://doi.org/10.1016/0012-365X(77)90044-9.