Related papers: Fluctuations of Subgraph Counts in Graphon Based R…
Our main results are quantitative bounds in the multivariate normal approximation of centred subgraph counts in random graphs generated by a general graphon and independent vertex labels. We are interested in these statistics because they…
We study the normal approximation of functionals of Poisson measures having the form of a finite sum of multiple integrals. When the integrands are nonnegative, our results yield necessary and sufficient conditions for central limit…
We study the point process $W$ in $\mathbb{R}^d$ obtained by adding an independent Gaussian vector to each point in $\mathbb{Z}^d$. Our main concern is the asymptotic size of fluctuations of the linear statistics in the large volume limit,…
L. Lov\'asz and B. Szegedy proved in 2006 that the limits of convergent graph sequences can be described by measurable symmetric functions $W: [0, 1]\times [0, 1]\to [0, 1]$ called graphons. In our present paper we investigate the structure…
We investigate the fluctuations of linear spectral statistics of a Wigner matrix $W\_N$ deformed by a deterministic diagonal perturbation $D\_N$, around a deterministic equivalent which can be expressed in terms of the free convolution…
Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the…
We explore in this paper sufficient conditions for the $H$-property to hold, with a particular focus on the so-called line graphons. A graphon is a symmetric, measurable function from the unit square $[0,1]^2$ to the closed interval…
Let $k \ge 3$ be a fixed integer. We exactly determine the asymptotic distribution of $\ln Z_k(G(n,m))$, where $Z_k(G(n,m))$ is the number of $k$-colourings of the random graph $G(n,m)$. A crucial observation to this aim is that the…
We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner-type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes…
We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $\gamma…
Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph $G(n,p)$. We use an idea of graph decompositions to give a…
Consider a random graph $G$ of size $N$ constructed according to a \textit{graphon} $w \, : \, [0,1]^{2} \mapsto [0,1]$ as follows. First embed $N$ vertices $V = \{v_1, v_2, \ldots, v_N\}$ into the interval $[0,1]$, then for each $i < j$…
One of the central topics in extremal graph theory is the study of the function $ex(n,H)$, which represents the maximum number of edges a graph with $n$ vertices can have while avoiding a fixed graph $H$ as a subgraph. Tur{\'a}n provided a…
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency…
We investigate the $H$-property for step-graphons. Specifically, we sample graphs $G_n$ on $n$ nodes from a step-graphon and evaluate the probability that $G_n$ has a Hamiltonian decomposition in the asymptotic regime as $n\to\infty$. It…
For a graph $G=(V,E)$ with $v(G)$ vertices the partition function of the random cluster model is defined by $$Z_G(q,w)=\sum_{A\subseteq E(G)}q^{k(A)}w^{|A|},$$ where $k(A)$ denotes the number of connected components of the graph $(V,A)$.…
We develop a theory of ultrametric graphons as limiting objects for random networks with nested hierarchical community structure. A graphon $W:[0,1]^2\to[0,1]$ is called ultrametric if $W(x,y)=w(d(x,y))$, where $d$ is an ultrametric on…
For given graphs $G$ and $H$, let $|Hom(G,H)|$ denote the set of graph homomorphisms from $G$ to $H$. We show that for any finite, $n$-regular, bipartite graph $G$ and any finite graph $H$ (perhaps with loops), $|Hom(G,H)|$ is maximum when…
Given a sequence of graphs $G_n$ and a fixed graph $H$, denote by $T(H, G_n)$ the number of monochromatic copies of the graph $H$ in a uniformly random $c$-coloring of the vertices of $G_n$. In this paper we study the joint distribution of…
For every fixed graph $H$ and every fixed $0 < \alpha < 1$, we show that if a graph $G$ has the property that all subsets of size $\alpha n$ contain the ``correct'' number of copies of $H$ one would expect to find in the random graph…