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Longest paths in random hypergraphs

Cooley, Oliver, Garbe, Frederik, Hng, Eng Keat, Kang, Mihyun, Sanhueza-Matama, Nicolás and Zalla, Julian (2021) Longest paths in random hypergraphs. SIAM Journal on Discrete Mathematics, 35 (4). 2430 – 2458. ISSN 0895-4801

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Identification Number: 10.1137/20M1345712

Abstract

The excellent target article of Hamm et al. (2022) raises much food for thought. In this commentary we first discuss what is included in their proposed category of ‘positive evaluations and responses to police assertions of power to attempt social influence’. Given integers k, j with 1 j k − 1, we consider the length of the longest j-tight path in the binomial random k-uniform hypergraph Hk(n, p). We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges. In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long j-tight path.

Item Type: Article
Official URL: https://epubs.siam.org/journal/sjdmec
Additional Information: © 2021 Society for Industrial and Applied Mathematics
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 13 Apr 2022 09:30
Last Modified: 15 Jun 2022 14:27
URI: http://eprints.lse.ac.uk/id/eprint/114877

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