Related papers: Eulerian magnitude homology: subgraph structure an…
We consider the random hypergraph on a finite vertex set by choosing each set of vertices as an hyperedge independently at random. We express the probability distributions of the (lower-)associated simplicial complex and the…
We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these…
We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…
This work addresses a modification of the random geometric graph (RGG) model by considering a set of points uniformly and independently distributed on the surface of a $(d-1)$-sphere with radius $r$ in a $d-$dimensional Euclidean space,…
The moduli space of rank $n$ graphs, the outer automorphism group of the free group of rank $n$ and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like…
We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy…
The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type $A$,…
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…
We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear…
A geometric entropy is defined as the Riemannian volume of the parameter space of a statistical manifold associated with a given network. As such it can be a good candidate for measuring networks complexity. Here we investigate its ability…
In this paper we introduce a new parameter for a graph called the {\it minimum universal rank}. This parameter is similar to the minimum rank of a graph. For a graph $G$ the minimum universal rank of $G$ is the minimum rank over all…
Magnitude homology of graphs is introduced by Hepworth and Willerton in arXiv:1505.04125 . Magnitude homology of arbitrary metric spaces by Leinster and Shulman in arXiv:1711.00802v2 . We verify that the K\"unneth and Mayer-Vietoris…
The aim of this paper is to present a few versions of the Riemann-Hurwitz formula for a regular branched covering of graphs. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first…
We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk…
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry…
The last decade has seen the development of path homology and magnitude homology -- two homology theories of directed graphs, each satisfying classic properties such as Kunneth and Mayer-Vietoris theorems. Recent work of Asao has shown that…
Often, high dimensional data lie close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds. The homology groups of a manifold are important topological invariants that provide an algebraic…
In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology…
Geometric scale-free random graphs are popular models for networks that exhibit as heavy-tailed degree distributions, small-worldness and high clustering. In these models, vertices have weights that cause the heavy-tailed degrees and are…
Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real…