Related papers: Large deviations for marked sparse random graphs w…
An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent set of size $C\log_2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very…
We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint…
It is well known, mainly because of the work of Kurtz, that density dependent Markov chains can be approximated by sets of ordinary differential equations (ODEs) when their indexing parameter grows very large. This approximation cannot…
Diffusion kernels over graphs have been widely utilized as effective tools in various applications due to their ability to accurately model the flow of information through nodes and edges. However, there is a notable gap in the literature…
A novel principle is presented which allows for the proof of bounded weak solutions to a class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure. The main feature of these systems is that…
Diameter, radius and eccentricities are fundamental graph parameters, which are extensively studied in various computational settings. Typically, computing approximate answers can be much more efficient compared with computing exact…
We propose a notion of graph convergence that interpolates between the Benjamini--Schramm convergence of bounded degree graphs and the dense graph convergence developed by L\'aszl\'o Lov\'asz and his coauthors. We prove that spectra of…
In this note, we give a precise description of the limiting empirical spectral distribution (ESD) for the non-backtracking matrices for an Erd\H{o}s-R\'{e}nyi graph assuming $np/\log n$ tends to infinity. We show that derandomizing part of…
Graph embedding provides a feasible methodology to conduct pattern classification for graph-structured data by mapping each data into the vectorial space. Various pioneering works are essentially coding method that concentrates on a…
We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk…
This paper investigates random walks and diffusion limits on a broad class of fractal graphs generated by Edge Iterated Graph Systems (EIGS). We prove that the rescaled simple random walks converge in the…
We propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process, and naturally generalizes existing probabilistic models with…
The Erd\H{o}s-R\'enyi random graph is the fundamental random graph model. In this paper we consider its continuous-time version, where multi-edges and self-loops are also allowed. It is well-known that the sizes of its connected components…
McKay proved that the limiting spectral measures of the ensembles of $d$-regular graphs with $N$ vertices converge to Kesten's measure as $N\to\infty$. In this paper we explore the case of weighted graphs. More precisely, given a large…
The objective of this paper is to study the characteristics (geometric and otherwise) of very large attribute based undirected networks. Real-world networks are often very large and fast evolving. Their analysis and understanding present a…
In recent decades, it has been emphasized that the evolving structure of networks may be shaped by interaction principles that yield sparse graphs with a vertex degree distribution exhibiting an algebraic tail, and other structural traits…
We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We…
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large…
We study the random loop model with crosses and bars on sparse random graphs. Our main objective is to prove the existence of macroscopic loops, in the sense that a loop visits a positive proportion of the vertices. We develop a…
We derive a large deviation principle for the empirical currents of lattice gas dynamics which combine a fast stirring mechanism (Symmetric Simple Exclusion Process) and creation/annihilation mechanisms (Glauber dynamics). Previous results…