Random Simplicial Complexes

Abstract

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 p ₀; on the next step we connect every pair of retained vertices by an edge with probability p ₁, and then fill in every triangle in the obtained random graph with probability p ₂, 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.