Related papers: Generalized Random Simplicial Complexes
We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters $p_\sigma$ approach neither $0$ nor $1$. We show that nontrivial Betti numbers…
The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate…
In this paper we introduce a method which allows us to study properties of the random uniform simplicial complex. That is, we assign equal probability to all simplicial complexes with a given number of vertices and then consider properties…
The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique…
The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of…
Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of…
Group interactions occur frequently in social settings, yet their properties beyond pairwise relationships in network models remain unexplored. In this work, we study homophily, the nearly ubiquitous phenomena wherein similar individuals…
We prove sharper versions of theorems of Linial-Meshulam and Meshulam-Wallach which describe the behavior for (Z/2)-cohomology of a random k-dimensional simplicial complex within a narrow transition window. In particular, we show that…
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the…
Simplicial complexes describe collaboration networks, protein interaction networks and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial…
Cascades arise in many contexts (e.g., neuronal avalanches, social contagions, and system failures). Despite evidence that propagations often involve higher-order dependencies, cascade theory has largely focused on models with…
Topological study of existing random simplicial complexes is non-trivial and has led to several seminal works. However, the applicability of such studies is limited since the randomness there is usually governed by a single parameter. With…
We survey techniques for constructing spaces with non-trivial self covers. These processes include methods for building low and high dimension continua which non-trivially self. We also discuss several related group theoretic and…
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
The Linial-Meshulam complex model is a natural higher-dimensional analog of the Erd\H{o}s-R\'enyi graph model. In recent years, Linial and Peled established a limit theorem for Betti numbers of Linial-Meshulam complexes with an appropriate…
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the…
A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links,…
We study a natural model of random 2-dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$-face is included independently with probability p. Our main result is to exhibit a sharp…
Many coupled problems in engineering and science can be described by elliptic partial differential equations on adjacent domains, where the coupling can be considered either as a thin equidimensional overlap between the model domains, or as…
We study $d$-dimensional simplicial complexes that are PL embeddable in $\mathbb{R}^{d+1}$. It is shown that such a complex must satisfy a certain homological condition. The existence of this obstruction allows us to provide a systematic…