Related papers: Explosive Percolation in Erd\"os-R\'enyi-Like Rand…
It is widely believed that certain simple modifications of the random graph process lead to discontinuous phase transitions. In particular, starting with the empty graph on $n$ vertices, suppose that at each step two pairs of vertices are…
Perhaps the best understood phase transition is that in the component structure of the uniform random graph process introduced by Erd\H{o}s and R\'enyi around 1960. Since the model is so fundamental, it is very interesting to know which…
The evolution of the usual Erd\H{o}s-R\'{e}nyi random graph model on n vertices can be described as follows: At time 0 start with the empty graph, with n vertices and no edges. Now at each time k, choose 2 vertices uniformly at random and…
A fundamental and very well studied region of the Erd\"os-R\'enyi process is the phase transition at n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine 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. Although the evolution of such `local'…
There is still much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph…
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…
Using the finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with minimum product of two connecting cluster sizes…
The Erd\H{o}s-R\'{e}nyi process begins with an empty graph on n vertices and edges are added randomly one at a time to a graph. A classical result of Erd\H{o}s and R\'{e}nyi states that the Erd\H{o}s-R\'{e}nyi process undergoes a phase…
The existence of explosive phase transitions in random (Erd\H os R\'enyi-type) networks has been recently documented by Achlioptas et al.\ [Science {\bf 323}, 1453 (2009)] via simulations. In this Letter we describe the underlying mechanism…
In an Achlioptas process, starting with a graph that has n vertices and no edge, in each round $d \geq 1$ edges are drawn uniformly at random, and using some rule exactly one of them is chosen and added to the evolving graph. For the class…
It has been recently shown that the percolation transition is discontinuous in Erd\H{o}s-R\'enyi networks and square lattices in two dimensions under the Achlioptas Process (AP). Here, we show that when the structure is highly heterogeneous…
Achlioptas processes are a class of dynamically grown random graphs where on each step several edges are chosen at random but only one is added. The sum rule, product rule, and bounded size rules have been extensively studied. Here we…
Percolation is one of the most studied processes in statistical physics. A recent paper by Achlioptas et al. [Science 323, 1453 (2009)] has shown that the percolation transition, which is usually continuous, becomes discontinuous…
We consider a natural variant of the Erd\H{o}s-R\'enyi random graph process in which $k$ vertices are special and are never put into the same connected component. The model is natural and interesting on its own, but is actually inspired by…
A significant generalization of the Erd\"os-R\'enyi random graph model is an `inhomogeneous' random graph where the edge probabilities vary according to vertex types. We identify the threshold value for this random graph with a finite…
We present a detailed study of the evolution of the giant component of the Erd\H{o}s-R\'enyi graph process as the mean degree increases from 1 to infinity. It leads to the identification of the limiting process of the rescaled fluctuations…
As suggested by Itai Benjamini, we introduced a variant of the Erd\"os- R\'enyi random graph process with a forbidden degree $k$, in which every edge adjacent to a vertex $v$ is removed when the degree of $v$ reaches $k$ (but the removed…
Achlioptas processes such as the Bohman--Frieze process are much harder to analyse than the classical Erd\H{o}s--R\'enyi process, due to the dependence between edges added at different stages. This dependence means that most analysis so far…
We study a one parameter family of random graph models that spans a continuum between traditional random graphs of the Erd\H{o}s-R\'enyi type, where there is no underlying structure, and percolation models, where the possible edges are…