Related papers: A logical approach to concentration
Let 0<p<1 be fixed. Shamir and Spencer proved in the 1980s that the chromatic number of a random graph in G(n,p) is concentrated in an interval of length about n^{1/2}. In this explanatory note, we give a proof of a result due due Noga…
Preferential attachment graphs are random graphs designed to mimic properties of typical real world networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already…
We study the behavior of the capital process of a continuous Bayesian mixture of fixed proportion betting strategies in the one-sided unbounded forecasting game in game-theoretic probability. We establish the relation between the rate of…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…
We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments.…
In the averaging process on a graph $G = (V, E)$, a random mass distribution $\eta$ on $V$ is repeatedly updated via transformations of the form $\eta_{v}, \eta_{w} \mapsto (\eta_{v} + \eta_{w})/2$, with updates made according to…
We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable…
Shelah Spencer [ShSp:304] proved the 0-1 law for the random graphs G(n,p_n), p_n=n^{- alpha}, alpha in (0,1) irrational (set of nodes in [n]= {1, ...,n}, the edges are drawn independently, probability of edge is p_n). One may wonder what…
We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality…
Similarity metrics are central in the theory of large networks and graph limits. For bounded-degree graphs, the Benjamini--Schramm metric records the distribution of rooted neighbourhoods, while the stronger colored-neighbourhood metric…
We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…
We develop a formalism to address statistical pattern recognition of graph valued data. Of particular interest is the case of all graphs having the same number of uniquely labeled vertices. When the vertex labels are latent, such graphs are…
Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters…
Statistical inference for exponential-family models of random graphs with dependent edges is challenging. We stress the importance of additional structure and show that additional structure facilitates statistical inference. A simple…
We consider general Exponential Random Graph Models (ERGMs) where the sufficient statistics are functions of homomorphism counts for a fixed collection of simple graphs $F_k$. Whereas previous work has shown a degeneracy phenomenon in dense…
Graph-structured data arise naturally in many different application domains. By representing data as graphs, we can capture entities (i.e., nodes) as well as their relationships (i.e., edges) with each other. Many useful insights can be…
Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that…
Given a graph $G$ and a constant $\gamma \in [0,1]$, let $\omega^{(\gamma)}(G)$ be the largest integer $r$ such that there exists an $r$-vertex subgraph of $G$ containing at least $\gamma \binom{r}{2}$ edges. It was recently shown that…
We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and…
Graph aggregation is the process of computing a single output graph that constitutes a good compromise between several input graphs, each provided by a different source. One needs to perform graph aggregation in a wide variety of…