Related papers: The homology of random simplicial complexes in the…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
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…
We investigate the asymptotic behavior, in the long time limit, of the random homology associated to realizations of stochastic diffusion processes on a compact Riemannian manifold. In particular a rigidity result is established: if the…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…
A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. In this paper, we study the embedded homology as well as the homology of the (lower-)associated simplicial complexes for hypergraphs. We…
This paper deals with the local asymptotic structure, in the sense of Le Cam's asymptotic theory of statistical experiments, of the signal detection problem in high dimension. More precisely, we consider the problem of testing the null…
In the seminal contribution [4] the joint weak convergence of maxima and minima of weakly dependent stationary sequences is derived under some mild asymptotic conditions. In this paper we address additionally the case of incomplete samples…
A central problem in topological data analysis is that of computing the homology of a given simplicial complex. Said complexes can have arbitrary large number of simplices, as can happen, for example, if the space is the Rips-Vietoris or…
For a simplicial complex $X$, the $d$-clique complex $\Delta_d(X)$ is the simplicial complex having all subsets of vertices whose $(d + 1)$-subsets are contained by $X$ as its faces. We prove that if $p = n^{\alpha}$, with $\alpha <…
Within the framework of generalized combinatorial approach, the complexity is determined for infinite set of self-similar hierarchical ensembles. This complexity is shown to increase with strengthening of the hierarchy coupling to the…
Deciding whether two simplicial complexes are homotopy equivalent is a fundamental problem in topology, which is famously undecidable. There exists a combinatorial refinement of this concept, called simple-homotopy equivalence: two…
We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the…
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…
Finding a cycle of lowest weight that represents a homology class in a simplicial complex is known as homology localization (HL). Here we address this NP-complete problem using parameterized complexity theory. We show that it is W[1]-hard…
In this article, we investigate the asymptotic properties of Bayesian multiple testing procedures under general dependent setup, when the sample size and the number of hypotheses both tend to infinity. Specifically, we investigate strong…
We provide dual algorithms for sampling the space of abstract simplicial complexes on a fixed number of vertices. We develop a generative and descriptive sampler designed with heuristics to help balance the combinatorial multiplicities 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.…
Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying…
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…
The asymptotic variance of the maximum likelihood estimate is proved to decrease when the maximization is restricted to a subspace that contains the true parameter value. Maximum likelihood estimation allows a systematic fitting of…