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Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex…
A temporal graph is a graph whose edges appear at certain points in time. These graphs are temporally connected (in class TC) if all vertices can reach each other by temporal paths (traversing the edges in chronological order). Reachability…
Higher-order connectivity patterns such as small induced sub-graphs called graphlets (network motifs) are vital to understand the important components (modules/functional units) governing the configuration and behavior of complex networks.…
In this paper we describe a triple correspondence between graph limits, information theory and group theory. We put forward a new graph limit concept called log-convergence that is closely connected to dense graph limits but its main…
This paper introduces a novel topology, referred to as the star topology, on finite graphs. By treating vertices and edges as points in a unified space, we explore continuous maps between Bare representations of a graph and their…
The tension equation for a mapping $f:{\mathbb C}\to {\mathbb C}$ is the nonlinear second order equation \[ \Delta f +\varphi(f) f_z f_{\bar z} = 0\] Solutions are "harmonic" mappings. Here we give a complete description of the solution…
How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use…
In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have achieved some…
This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the…
Mapping class groups of locally finite graphs are the analogue of those of infinite-type surfaces, and serve as a "big" version of $\text{Out}(F_n)$. In this paper, we investigate which of these mapping class groups have a dense conjugacy…
Motivated by applications to a wide range of assemble-to-order systems, operations scheduling, healthcare systems and collaborative economy applications, we introduce a stochastic matching model on hypergraphs, extending the model in [15]…
This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II…
A graph homomorphism between two graphs is a map from the vertex set of one graph to the vertex set of the other graph, that maps edges to edges. In this note we study the range of a uniformly chosen homomorphism from a graph G to the…
We give elementary applications of quasi-homomorphisms to growth problems in groups. A particular case concerns the number of torsion elements required to factorise a given element in the mapping class group of a surface.
In this paper, we study the behaviour of TF-isomorphisms, a natural generalisation of isomorphisms. TF-isomorphisms allow us to simplify the approach to seemingly unrelated problems. In particular, we mention the Neighbourhood…
We study topological groups of monotonic autohomeomorphisms on a generalized ordered space $L$. We find a condition that is necessary and sufficient for the set of all monotonic autohomeomorphisms on $L$ along with the function composition…
Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for…
For a continuous self-map $T$ of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given…
We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with…
Torsion and Betti numbers for knots are special cases of more general invariants associated to a finitely generated group G and epimorphism from G to the integers. The sequence of Betti numbers is always periodic; under mild hypotheses, the…