Related papers: Sparse graphs and their Benjamini-Schramm limits: …
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs…
We study the effect of the choice of embedding geometry on the entropy of random geometric graph ensembles with soft connection functions. First we show that when the connection range is small, the entropy is dependent only on the dimension…
We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we…
We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A…
Expanders are sparse graph that are strongly connected, where {\it connectivity} is quantified using eigenvalues of the adjacency matrix, and {\it sparsity} in terms of vertex valency. We give a model of random graphs and study their…
We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows to derive a series of spectral…
There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph…
Random $s$-intersection graphs have recently received considerable attention in a wide range of application areas. In such a graph, each vertex is equipped with a set of items in some random manner, and any two vertices establish an…
Let a graph be observed through a finite random sampling mechanism. Spectral methods are routinely applied to such graphs, yet their outputs are treated as deterministic objects. This paper develops finite-sample inference for spectral…
In this paper we consider alignment of sparse graphs, for which we introduce the Neighborhood Tree Matching Algorithm (NTMA). For correlated Erd\H{o}s-R\'{e}nyi random graphs, we prove that the algorithm returns -- in polynomial time -- a…
The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and…
A flip of a graph is obtained by complementing the edge relation within a set of vertices. Flips are typically used to separate vertices in a graph, by increasing the distances between them. We show that in $K_{t,t}$-free graphs, every…
Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree.…
We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global…
Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…
For graphs $G$ and $H$, let $G\to H$ signify that any red/blue edge coloring of $G$ contains a monochromatic $H$. Let $G(N,p)$ be the random graph of order $N$ and edge probability $p$. The Ramsey thresholds for fixed graphs have received…
The notion of barycentre graph is of crucial importance for machine learning algorithms that process graph-valued data. The barycentre graph is a "summary graph" that captures the mean topology and connectivity structure of a training…
We discuss relations between the amenability of a graph and spectral properties of a random walk driven by a dynamical system. In order to include graphs which are not locally compact, we introduce the concept of amenability of weighted…
Spontaneous symmetry breaking is a cornerstone of modern physics, defining a wealth of phenomena in condensed-matter and high-energy physics, and beyond. It requires an infinite number of degrees of freedom, and even then, for continuous…
For various purposes and, in particular, in the context of data compression, a graph can be examined at three levels. Its structure can be described as the unlabeled version of the graph; then the labeling of its structure can be added; and…