Related papers: The giant in random graphs is almost local

We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form…

We study properties of Graph Convolutional Networks (GCNs) by analyzing their behavior on standard models of random graphs, where nodes are represented by random latent variables and edges are drawn according to a similarity kernel. This…

Specify a randomized algorithm that, given a very large graph or network, extracts a random subgraph. What can we learn about the input graph from a single subsample? We derive laws of large numbers for the sampler output, by relating…

PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called `power-law…

For a graph $G$, we denote by $\alpha(G)$ the size of a maximum independent set and by $\omega(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $\alpha$ and $\omega$, respectively. One…

We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and its…

We study Gibbs partitions that typically form a unique giant component. The remainder is shown to converge in total variation toward a Boltzmann-distributed limit structure. We demon- strate how this setting encompasses arbitrary weighted…

We define direct sums and a corresponding notion of connectedness for graph limits. Every graph limit has a unique decomposition as a direct sum of connected components. As is well-known, graph limits may be represented by symmetric…

A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs…

We consider bond percolation on random graphs with given degrees and bounded average degree. In particular, we consider the order of the largest component after the random deletion of the edges of such a random graph. We give a rough…

We provide a complete description of the giant component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as soon as it emerges from the scaling window, i.e., for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to \infty$ and $\epsilon=o(1)$. Our…

In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+\epsilon)/n$ where $\epsilon^3 n…

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time…

We consider the component structure of a recent model of random graphs on the hyperbolic plane that was introduced by Krioukov et al. The model exhibits a power law degree sequence, small distances and clustering, features that are…

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…

A temporal graph is a graph whose edges appear only at certain points in time. Recently, the second and the last three authors proposed a natural temporal analog of the Erd\H{o}s-R\'enyi random graph model. The proposed model is obtained by…

We study the component structure of the random graph $G=G_{n,m,d}$. Here $d=O(1)$ and $G$ is sampled uniformly from ${\mathcal G}_{n,m,d}$, the set of graphs with vertex set $[n]$, $m$ edges and maximum degree at most $d$. If $m=\mu n/2$…

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p)…

Consider a 2-dimensional soft random geometric graph $G(\lambda,s,\phi)$, obtained by placing a Poisson($\lambda s^2$) number of vertices uniformly at random in a square of side $s$, with edges placed between each pair $x,y$ of vertices…

The poster presents an analytic formalism describing metric properties of undirected random graphs with arbitrary degree distributions and statistically uncorrelated (i.e. randomly connected) vertices. The formalism allows to calculate the…