Related papers: Random simplicial complexes
In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special…
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…
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 expository article, we survey the rapidly emerging area of random geometric simplicial complexes.
There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a…
We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the…
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…
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,…
Spin (spherical) random fields are very important in many physical applications, in particular they play a key role in Cosmology, especially in connection with the analysis of the Cosmic Microwave Background radiation. These objects can be…
We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs.…
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…
We review models of random geometries based on the dynamical lattice approach. We discuss one dimensional model of simplicial complexes (branched polymers), two dimensional model of dynamical triangulations and four dimensional model of…
It is increasingly common for data to possess intricate structure, necessitating new models and analytical tools. Graphs, a prominent type of structure, can encode the relationships between any two entities (nodes). However, graphs neither…
Random walks on a graph reflect many of its topological and spectral properties, such as connectedness, bipartiteness and spectral gap magnitude. In the first part of this paper we define a stochastic process on simplicial complexes of…
Given a chain complex with the only modification that each cell of the complex has a probability distribution assigned. We will call this complex - a random complex and what should be understood in practice, is that we have a classical…
In our recent work we described conditions under which a multi-parameter random simplicial complex is connected and simply connected. We showed that the Betti numbers of multi-parameter random simplicial complexes in one specific dimension…
We consider a multi-parameter model for randomly constructing simplicial complexes. This model interpolates between random clique complexes and Linial-Meshulam random $k$-dimensional complexes, two models that have been extensively studied.…
We define a class of probability distributions that we call simplicial mixture models, inspired by simplicial complexes from algebraic topology. The parameters of these distributions represent their topology and we show that it is possible…
We provide a short introduction to the field of topological data analysis and discuss its possible relevance for the study of complex systems. Topological data analysis provides a set of tools to characterise the shape of data, in terms of…
A general method for constructing simplicial complex from observed time series of dynamical systems based on the delay coordinate reconstruction procedure is presented. The obtained simplicial complex preserves all pertinent topological…