The Phase Transition Concerning the Giant Component in a Sparse Random Graph: A Theorem of Erdős and Rényi

2014 ◽  
pp. 109-131
Author(s):  
Ross G. Pinsky
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Giant Component

scholarly journals The Phase Transition in the Configuration Model

2012 ◽  
Vol 21 (1-2) ◽  
pp. 265-299 ◽  
Author(s):  
OLIVER RIORDAN
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Regular Graph ◽  
Degree Sequence ◽  
Natural Extension ◽  
Giant Component ◽  
Configuration Model ◽  
Random Subgraph ◽  
Single Method ◽  
Present Setting

Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r ≥ 3 is fixed and n = |G| → ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the ‘scaling window’ of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobás and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.

scholarly journals A Simple Branching Process Approach to the Phase Transition in $G_{n,p}$

10.37236/2588 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Béla Bollobás ◽  
Oliver Riordan
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Branching Process ◽  
Process Approach ◽  
Giant Component ◽  
Asymptotic Size ◽  
The Way

It is well known that the branching process approach to the study of the random graph $G_{n,p}$ gives a very simple way of understanding the size of the giant component when it is fairly large (of order $\Theta(n)$). Here we show that a variant of this approach works all the way down to the phase transition: we use branching process arguments to give a simple new derivation of the asymptotic size of the largest component whenever $(np-1)^3n\to\infty$.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

The cover time of the giant component of a random graph

2008 ◽  
Vol 32 (4) ◽  
pp. 401-439 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Giant Component ◽  
Cover Time

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

scholarly journals Convergence of Achlioptas Processes via Differential Equations with Unique Solutions

2015 ◽  
Vol 25 (1) ◽  
pp. 154-171 ◽  
Author(s):  
OLIVER RIORDAN ◽  
LUTZ WARNKE
Keyword(s):  
Differential Equations ◽  
Stochastic Processes ◽  
Large Class ◽  
Unique Solution ◽  
Random Graph ◽  
Giant Component ◽  
General Idea ◽  
Product Rule ◽  
System Of Differential Equations ◽  
Empty 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 largest component in such variations of the Erdős–Rényi random graph process has recently received considerable attention, in particular for Bollobás's ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes.Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.

Components of Random Forests

1992 ◽  
Vol 1 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Tomasz Łuczak ◽  
Boris Pittel
Keyword(s):  
Phase Transition ◽  
Random Forest ◽  
Asymptotic Behaviour ◽  
Random Graph ◽  
Random Forests ◽  
Limit Distribution ◽  
The Family ◽  
Supercritical Phase

A forest ℱ(n, M) chosen uniformly from the family of all labelled unrooted forests with n vertices and M edges is studied. We show that, like the Érdős-Rényi random graph G(n, M), the random forest exhibits three modes of asymptotic behaviour: subcritical, nearcritical and supercritical, with the phase transition at the point M = n/2. For each of the phases, we determine the limit distribution of the size of the k-th largest component of ℱ(n, M). The similarity to the random graph is far from being complete. For instance, in the supercritical phase, the giant tree in ℱ(n, M) grows roughly two times slower than the largest component of G(n, M) and the second largest tree in ℱ(n, M) is of the order n⅔ for every M = n/2 +s, provided that s3n−2 → ∞ and s = o(n), while its counterpart in G(n, M) is of the order n2s−2 log(s3n−2) ≪ n⅔.

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