A random triadic process

Dániel Korándi; Yuval Peled; Benny Sudakov

doi:10.1137/15M1012487

A random triadic process

Dániel Korándi, Yuval Peled, Benny Sudakov

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G₀ on the same vertex set, containing all the edges incident to some vertex v₀, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy∈ H. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is p = 1/2√n. We conclude that p = 1 /2√n is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.

Original language	American English
Pages (from-to)	1-19
Number of pages	19
Journal	SIAM Journal on Discrete Mathematics
Volume	30
Issue number	1
DOIs	https://doi.org/10.1137/15M1012487
State	Published - 2016

Bibliographical note

Funding Information:
Email: yuvalp@cs.huji.ac.il. Yuval Peled is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. 3 Email: benjamin.sudakov@math.ethz.ch. Research supported in part by SNSF grant 200021-149111.

Publisher Copyright:
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Keywords

Differential equation method
Propagation
Random simplicial complexes
Triadic process

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10.1137/15M1012487

Cite this

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title = "A random triadic process",

abstract = "Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G0 on the same vertex set, containing all the edges incident to some vertex v0, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy∈ H. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is p = 1/2√n. We conclude that p = 1 /2√n is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.",

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doi = "10.1137/15M1012487",

language = "American English",

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AU - Korándi, Dániel

AU - Peled, Yuval

AU - Sudakov, Benny

N1 - Funding Information: Email: yuvalp@cs.huji.ac.il. Yuval Peled is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. 3 Email: benjamin.sudakov@math.ethz.ch. Research supported in part by SNSF grant 200021-149111. Publisher Copyright: Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

PY - 2016

Y1 - 2016

N2 - Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G0 on the same vertex set, containing all the edges incident to some vertex v0, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy∈ H. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is p = 1/2√n. We conclude that p = 1 /2√n is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.

AB - Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G0 on the same vertex set, containing all the edges incident to some vertex v0, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy∈ H. We say that the process propagates if it reaches the complete graph before it terminates. In this paper we prove that the threshold probability for propagation is p = 1/2√n. We conclude that p = 1 /2√n is an upper bound for the threshold probability that a random 2-dimensional simplicial complex is simply connected.

KW - Differential equation method

KW - Propagation

KW - Random simplicial complexes

KW - Triadic process

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JO - SIAM Journal on Discrete Mathematics

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IS - 1

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A random triadic process

Abstract

Bibliographical note

Keywords

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