相关论文: Distribution of diameters for Erd\"os-R\'enyi rand…

We consider the problem of determining the proportion of edges that are discovered in an Erdos-Renyi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of…

We study the large-deviation properties of minimum spanning trees for two ensembles of random graphs with $N$ nodes. First, we consider complete graphs. Second, we study Erd\H{o}s-R\'{e}nyi (ER) random graphs with edge probability $p=c/N$…

In this paper we explore maximal deviations of large random structures from their typical behavior. We introduce a model for a high-dimensional random graph process and ask analogous questions to those of Vapnik and Chervonenkis for…

We study the size of the largest biconnected components in sparse Erd\H{o}s-R\'enyi graphs with finite connectivity and Barab\'asi-Albert graphs with non-integer mean degree. Using a statistical-mechanics inspired Monte Carlo approach we…

The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, 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 study the near-critical behavior of the sparse Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$ on $n\gg1$ vertices, where the connection probability $p$ satisfies $np = 1+\theta(b_n^2/n)^{1/3}$, with $n^{3/10}\ll {b_n}\ll n^{1/2}$, and…

In the random geometric graph model $\mathsf{Geo}_d(n,p)$, we identify each of our $n$ vertices with an independently and uniformly sampled vector from the $d$-dimensional unit sphere, and we connect pairs of vertices whose vectors are…

In this paper, we consider the Ising model on random $d$-regular graphs (with $d\ge3$) and Erd\"os-R\'enyi graphs $G(n,d/n)$ (with $d>1$) at the critical temperature. We prove that the \textit{magnetization}, i.e.\ the sum of the spins of a…

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erd\H{o}s-R\'enyi graph with parameter $p_n\in (0, 1]$,…

In dense Erd\H{o}s-R\'enyi random graphs, we are interested in the events where large numbers of a given subgraph occur. The mean behavior of subgraph counts is known, and only recently were the related large deviations results discovered.…

Consider the problem of determining the maximal induced subgraph in a random $d$-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large $d$, that any such…

This paper concerns the large deviations of a system of interacting particles on a random graph. There is no stochasticity, and the only sources of disorder are the random graph connections, and the initial condition. The average number of…

In this paper we study the diameter of the random graph $G(n,p)$, i.e., the the largest finite distance between two vertices, for a wide range of functions $p=p(n)$. For $p=\la/n$ with $\la>1$ constant, we give a simple proof of an…

Consider a critical Erd\"os-R\'enyi random graph: $n$ is the number of vertices, each one of the $\binom{n}{2}$ possible edges is kept in the graph independently from the others with probability $n^{-1}+\lambda n^{-4/3}$, $\lambda$ being a…

It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately…

We consider an Erdos-Renyi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N->infinity one obtains a graph of finite mean degree c. In…

The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…

We consider the random geometric graph on $n$ vertices drawn uniformly from a $d$--dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of $d$ and $n$. We show that, when $d$ is larger than $n$…

We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval…