Related papers: Simplicial Complex Emergence on Directed Hypergrap…
Over the past two decades, complex network theory provided the ideal framework for investigating the intimate relationships between the topological properties characterizing the wiring of connections among a system's unitary components and…
The singular simplicial set Sing(X) of a space X completely captures its weak homotopy type. We introduce a category of_controlled sets_, yielding _simplicial controlled sets_, such that one can functorially produce a singular simplicial…
Dependency networks (Heckerman et al., 2000) provide a flexible framework for modeling complex systems with many variables by combining independently learned local conditional distributions through pseudo-Gibbs sampling. Despite their…
Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial…
Disease spreading models such as the ubiquitous SIS compartmental model and its numerous variants are widely used to understand and predict the behaviour of a given epidemic or information diffusion process. A common approach to imbue more…
This article concerns properties of mixed $\ell$-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of…
Computation of persistent homology of simplicial representations such as the Rips and the C\v{e}ch complexes do not efficiently scale to large point clouds. It is, therefore, meaningful to devise approximate representations and evaluate the…
We show how the framework of crossed simplicial groups may be used to provide a classification of topological field theories on open cobordism categories defined by reductions of the structure group to a planar Lie group. Such theories are…
High order networks are weighted hypergraphs col- lecting relationships between elements of tuples, not necessarily pairs. Valid metric distances between high order networks have been defined but they are difficult to compute when the…
Adaptive social networks, in which nodes and network structure co-evolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on…
In this paper we develop a framework to study observability for uniform hypergraphs. Hypergraphs, being extensions of graphs, allow edges to connect multiple nodes and unambiguously represent multi-way relationships which are ubiquitous in…
Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological…
We present a new model which represents data as a mixture of simplices. Simplices are geometric structures that generalize triangles. We give a simple geometric understanding that allows us to learn a simplicial structure efficiently. Our…
It has been proposed that adaptation in complex systems is optimized at the critical boundary between ordered and disordered dynamical regimes. Here, we review models of evolving dynamical networks that lead to self-organization of network…
In this paper we investigate phenomena of spontaneous emergence or purposeful formation of highly organized structures in networks of related agents. We show that the formation of large organized structures requires exponentially large, in…
A complex system with many interacting individuals can be represented by a network consisting of nodes and links representing individuals and pairwise interactions, respectively. However, real-world systems grow with time and include many…
We consider the population dynamics of a set of species whose network of catalytic interactions is described by a directed graph. The relationship between the attractors of this dynamics and the underlying graph theoretic structures like…
We study norm-based uniform convergence bounds for neural networks, aiming at a tight understanding of how these are affected by the architecture and type of norm constraint, for the simple class of scalar-valued one-hidden-layer networks,…
The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the…
Complex networks are the subject of fundamental interest from the scientific community at large. Several metrics have been introduced to characterize the structure of these networks, such as the degree distribution, degree correlation, path…