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When does the top homology of a random simplicial complex vanish?
Author(s) -
Aronshtam Lior,
Linial Nathan
Publication year - 2015
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20495
Subject(s) - vertex (graph theory) , mathematics , combinatorics , upper and lower bounds , simplicial complex , homology (biology) , simplicial homology , struct , persistent homology , random graph , discrete mathematics , computer science , graph , algorithm , mathematical analysis , amino acid , biology , biochemistry , programming language
Several years ago Linial and Meshulam (Combinatorica 26 (2006) 457–487) introduced a model calledX d ( n , p ) of random n ‐vertex d ‐dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability p = p ( n ) at which the d ‐dimensional homology of such a random d ‐complex is, almost surely, nonzero? Here we derive an upper bound on this threshold. Computer experiments that we have conducted suggest that this bound may coincide with the actual threshold, but this remains an open question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 26–35, 2015