Related papers: Coboundary expanders
Generalised degrees provide a natural bridge between local and global topological properties of networks. We define the generalised degree to be the number of neighbours of a node within one and two steps respectively. Tailored random graph…
A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs…
We study fundamental groups of clique complexes associated to random graphs. We establish thresholds for their cohomological and geometric dimension and torsion. We also show that in certain regime any aspherical subcomplex of a random…
For vertex and edge connectivity we construct infinitely many pairs of regular graphs with the same spectrum, but with different connectivity.
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group…
In this article we consider several forms of expansivity. We introduce two new definitions related with topological dimension. We study the topology of local stable sets under cw-expansive surface homeomorphisms and expansive homeomorphisms…
We formulate the geometric P=W conjecture for singular character varieties. We establish it for compact Riemann surfaces of genus one, and obtain partial results in arbitrary genus. To this end, we employ non-Archimedean, birational and…
This paper aims at studying the sample complexity of graph convolutional networks (GCNs), by providing tight upper bounds of Rademacher complexity for GCN models with a single hidden layer. Under regularity conditions, theses derived…
We prove Farber's conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our…
In this paper we develop an axiomatic approach to coarse homology theories. We prove a uniqueness result concerning coarse homology theories on the category of `coarse CW-complexes'. This uniqueness result is used to prove a version of the…
We prove that random triangulations of high genus contain very large expander subgraphs, answering a question of Benjamini. Our approach relies on new general criteria for arbitrary graphs to contain large expander subgraphs.
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly…
We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree…
In this paper, we consider a weighted generalization of the chromatic number of a Binomial random graph~\(G.\) We equip each edge with a random weight and then colour the vertices in such a way that the absolute colour difference between…
Networks are mathematical structures that are universally used to describe a large variety of complex systems such as the brain or the Internet. Characterizing the geometrical properties of these networks has become increasingly relevant…
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph…
In this paper, we consider a variation on Cheeger numbers related to the coboundary expanders recently defined by Dotterer and Kahle. A Cheeger-type inequality is proved, which is similar to a result on graphs due to Fan Chung. This…
We prove upper bounds on the face numbers of simplicial complexes in terms on their girths, in analogy with the Moore bound from graph theory. Our definition of girth generalizes the usual definition for graphs.
We consider a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary…
In an increasingly interconnected world, understanding and summarizing the structure of these networks becomes increasingly relevant. However, this task is nontrivial; proposed summary statistics are as diverse as the networks they…