Related papers: On two biased graph processes

Preferential attachment networks with power law exponent $\tau>3$ are known to exhibit a phase transition. There is a value $\rho_{\rm c}>0$ such that, for small edge densities $\rho\leq \rho_c$ every component of the graph comprises an…

We investigate the genus $g(n,m)$ of the Erd\H{o}s-R\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$…

Stochastic Kronecker graphs are a model for complex networks where each edge is present independently according the Kronecker (tensor) product of a fixed matrix k-by-k matrix P with entries in [0,1]. We develop a novel correspondence…

In the classical Erd\"os-R\'enyi random graph G(n,p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n,p) is homogeneous in the sense that all vertices have the same…

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn,…

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while…

We study the phase transition of the random degree constrained process (RDCP), a time-evolving random graph model introduced by Ruci\'nski and Wormald that generalizes the random $d$-process to the non-regular setting: each vertex of the…

In this work, we study some statistical properties of the extreme eigenstates of the randomly-weighted adjacency matrices of random graphs. We focus on two random graph models: Erd\H{o}s-R\'{e}nyi (ER) graphs and random geometric graphs…

We investigate the SIR epidemic on a dynamic inhomogeneous Erd\H{o}s-R\'enyi random graph, in which vertices are of one of $k$ types and in which edges appear and disappear independently of each other. We establish a functional law of large…

We consider the problem of estimating the edge density of densest $K$-node subgraphs of an Erd\"os-R\'{e}nyi graph $\mathbb{G}(n,1/2)$. The problem is well-understood in the regime $K=\Theta(\log n)$ and in the regime $K=\Theta(n)$. In the…

For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected…

We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a…

We study properties of some standard network models when the population is split into two types and the connection pattern between the types is varied. The studied models are generalizations of the Erd\H{o}s-R\'{e}nyi graph, the…

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the…

We study the algorithmic problem of finding a large independent set in the Erd{\"o}s-R\'{e}nyi random graph $G(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2\log_b n$, while a simple greedy algorithm -…

A sequence $D = \{d_1,...d_n\}$ is a feasible degree sequence if there is a graph on $\{1,...,n\}$ such that $i$ has degree $d_i$. For such a sequence, $G(D)$ is a graph chosen uniformly at random from those with the given degree sequence.…

In the inhomogeneous random graph model, each vertex $i\in\{1,\ldots,n\}$ is assigned a weight $W_i\sim\text{Unif}(0,1)$, and an edge between any two vertices $i,j$ is present with probability $k(W_i,W_j)/\lambda_n\in[0,1]$, where $k$ is a…

Given a graph $H$, we say that a graph $G$ is $H$-saturated if $G$ contains no copy of $H$ but adding any new edge to $G$ creates a copy of $H$. Let $sat(n,K_r,t)$ be the minimum number of edges in a $K_r$-saturated graph on $n$ vertices…

The size of a largest independent set of vertices in a given graph $G$ is denoted by $\alpha(G)$ and is called its independence number (or stability number). Given a graph $G$ and an integer $K,$ it is NP-complete to decide whether…

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward $p$-stars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph…