Related papers: Large random simplicial complexes, I
A wide variety of complex systems are characterized by interactions of different types involving varying numbers of units. Multiplex hypergraphs serve as a tool to describe such structures, capturing distinct types of higher-order…
We describe a framework for estimating Hilbert-Samuel multiplicities $e_XY$ for pairs of projective varieties $X \subset Y$ from finite point samples rather than defining equations. The first step involves proving that this multiplicity…
The intersection matrix of a finite simplicial complex has as each of its entries the rank of the intersection of its respective simplices. We prove that such matrix defines the triangulation of a closed connected surface up to isomorphism.
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to…
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…
The spectra of random feature matrices provide essential information on the conditioning of the linear system used in random feature regression problems and are thus connected to the consistency and generalization of random feature models.…
This article proposes a bivariate Simplex distribution for modeling continuous outcomes constrained to the interval $(0,1)$, which can represent proportions, rates, or indices. We derive analytical expressions to calculate the dependence…
We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely…
We introduce a novel percolation model that generalizes the classical Random Connection Model (RCM) to a random simplicial complex, allowing for a more refined understanding of connectivity and emergence of large-scale structures in random…
Intersection homology is defined for simplicial, singular and PL chains and it is well known that the three versions are isomorphic for a full filtered simplicial complex. In the literature, the isomorphism, between the singular and the…
All interesting and fascinating collective properties of a complex system arise from the intricate way in which its components interact. Various systems in physics, biology, social sciences and engineering have been successfully modelled as…
The task of this survey is to present various results on intersection patterns of convex sets. One of main tools for studying intersection patterns is a point of view via simplicial complexes. We recall the definitions of so called…
If only limited control over a multiparticle quantum system is available, a viable method to characterize correlations is to perform random measurements and consider the moments of the resulting probability distribution. We present…
We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…
Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R^{2n+1} necessarily exhibit complicated linking behaviour, thereby extending known…
We develop simple methods for constructing likelihoods and parameter priors for learning about the parameters and structure of a Bayesian network. In particular, we introduce several assumptions that permit the construction of likelihoods…
We extend some sharp inequalities for martingale-differences to general multiplicative systems of random variables. The key ingredient in the proofs is a technique reducing the general case to the case of Rademacher random variables without…
A model of rank polysemantic distribution with a minimal number of fitting parameters is offered. In an ideal case a parameter-free description of the dependence on the basis of one or several immediate features of the distribution is…
We study the problem of multiple hypothesis testing for multidimensional data when inter-correlations are present. The problem of multiple comparisons is common in many applications. When the data is multivariate and correlated, existing…
We extend the classical preferential attachment random graph model to random simplicial complexes. At each stage of the model, we choose one of the existing $k$-simplices with probability proportional to its $k$-degree. The chosen…