Random Simplicial Complexes
By Armindo Costa, Michael Farber. 2014
83 citations recorded by the source corpus.
In this paper we propose a model of random simplicial complexes with randomness in all dimensions. We start with a set of n vertices and retain each of them with probability p0; on the next step we connect every pair of retained vertices by an edge with probability p1, and then fill in every triangle in the obtained random graph with probability p2, and so on. As the result we obtain a random simplicial complex depending on the set of probability parameters (\(p_{0},p_{1},\ldots,p_{r}\)), 0 ≤ p i ≤ 1. The multi-parameter random simplicial complex includes both Linial-Meshulam and random clique complexes as special cases. Topological and geometric properties of this random simplicial complex depend on the whole set of parameters and their thresholds can be understood as convex subsets and not as single numbers as in all the previously studied models. We mainly focus on foundations and on containment properties of our multi-parameter random simplicial complexes. One may associate to any finite simplicial complex S a reduced density domain \(\tilde{\mu }(S) \subset \mathbf{R}^{r}\) (a convex domain) which fully controls information about the values of the multi-parameter for which the random complex contains S as a simplicial subcomplex. We also analyse balanced simplicial complexes and give positive and negative examples. We apply these results to describe dimension of a random simplicial complex.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Physiology
View Paper: https://www.semanticscholar.org/paper/1bfbca3e71c60a7c53c1662195bf25ddc5dd77ff
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Citations in the corpus
- Simplicial models of social contagion (2018, background): “While these yield Erd¨os-R´enyi-like models 54,55 of arbitrary complexity, here we are interested in models generating simplicial complexes with simplices of different dimension in which we can control and tune the expected local connectivity, e.g. the number of edges and “full” triangles a node…”
- Large random simplicial complexes, I (2015, methodology): “y two vertices are connected by an edge). Compared with the Linial - Meshulam model, the clique complex has randomness in dimension one but it influences its structure in all the higher dimensions. In [7] we initiated the study of a more general and more flexible model of random simplicial complexes w”