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.
Bibliographical noteFunding Information:
Email: email@example.com. Yuval Peled is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship. 3 Email: firstname.lastname@example.org. Research supported in part by SNSF grant 200021-149111.
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- Differential equation method
- Random simplicial complexes
- Triadic process