Related papers: Stability of Hypergraph Invariants and Transformat…

We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds,…

Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…

We introduce a taxonomy of interaction types and show that graphs are focal hypergraphs: every graph is canonically a focal hypergraph via its closed neighbourhood structure, and every graph dynamical model is a special case of the general…

Hypergraphs have emerged as a powerful modeling framework to represent systems with multiway interactions, that is systems where interactions may involve an arbitrary number of agents. Here we explore the properties of real-world…

In ``Characterization, stability and convergence of hierarchical clustering methods'' by G. E. Carlsson, F. Memoli, the natural way to construct an ultrametric space from a given metric space was presented. It was shown that the…

We prove that the reconstruction of a certain type of length spaces from their travel time data on a closed subset is Lipschitz stable. The travel time data is the set of distance functions from the entire space, measured on the chosen…

We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…

We study statistical and algorithmic aspects of using hypergraphons, that are limits of large hypergraphs, for modeling higher-order interactions. Although hypergraphons are extremely powerful from a modeling perspective, we consider a…

Hypergraphs naturally represent group interactions, which are omnipresent in many domains: collaborations of researchers, co-purchases of items, joint interactions of proteins, to name a few. In this work, we propose tools for answering the…

Graph embeddings have emerged as a powerful tool for representing complex network structures in a low-dimensional space, enabling the use of efficient methods that employ the metric structure in the embedding space as a proxy for the…

Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…

The primary objective of this paper is to introduce Hyers-Ulam-type stability results for monotone, subadditive, and convex graphs. We consider their standard definitions in an approximate sense and demonstrate the existence of a…

We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with one-dimensional concave fibre maps under multiplicative forcing when…

Graph similarity metrics serve far-ranging purposes across many domains in data science. As graph datasets grow in size, scientists need comparative tools that capture meaningful differences, yet are lightweight and scalable. Graph Relative…

In this paper we obtain an almost sure invariance principle for convergent sequences of either Anosov diffeomorphisms or expanding maps on compact Riemannian manifolds and prove an ergodic stability result for such sequences. The sequences…

We derive mean-field information Hessian matrices on finite graphs. The "information" refers to entropy functions on the probability simplex. And the "mean-field" means nonlinear weight functions of probabilities supported on graphs. These…

Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the…

From longitudinal biomedical studies to social networks, graphs have emerged as a powerful framework for describing evolving interactions between agents in complex systems. In such studies, after pre-processing, the data can be represented…

We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…

Metric graphs are special types of metric spaces used to model and represent simple, ubiquitous, geometric relations in data such as biological networks, social networks, and road networks. We are interested in giving a qualitative…