Related papers: Eulerian magnitude homology: subgraph structure an…
We equip the edges of a deterministic graph $H$ with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in $H$ satisfying the property that…
Graph homologies are powerful tools to compute the rational homotopy group of the space of long embeddings. Two graph homologies have been invented from two approaches to study the space of long embeddings: the hairy graph homology from…
We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes…
Force-directed layouts belong to a popular class of methods used to position nodes in a node-link diagram. However, they typically lack direct consideration of global structures, which can result in visual clutter and the overlap of…
For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…
We associate all small subgraph counting problems with a systematic graph encoding/representation system which makes a coherent use of graphlet structures. The system can serve as a unified foundation for studying and connecting many…
We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of nodes representing vertices and a set of nodes representing hyper-edges, with two nodes being adjacent if the corresponding vertex belongs to the corresponding…
Many inequality relations between real vector quantities can be succinctly expressed as "weak (sub)majorization" relations. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian…
Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these…
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…
We introduce new methods for understanding the topology of $\Hom$ complexes (spaces of homomorphisms between two graphs), mostly in the context of group actions on graphs and posets. We view $\Hom(T,-)$ and $\Hom(-,G)$ as functors from…
Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high dimensional anologues of spaces of long knots can be calculated as the homology of a direct sum of finite…
We introduce submodular hypergraphs, a family of hypergraphs that have different submodular weights associated with different cuts of hyperedges. Submodular hypergraphs arise in clustering applications in which higher-order structures carry…
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…
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…
Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…
Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the…
In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…
We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the…
We present a new algorithm for computing the first discrete homology group of a graph. By testing the algorithm on different data sets of random graphs, we find that it significantly outperforms other known algorithms.