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.

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

scholarly journals Phase transition on the degree sequence of a random graph process with vertex copying and deletion

2011 ◽  
Vol 121 (4) ◽  
pp. 885-895 ◽  
Author(s):  
Kai-Yuan Cai ◽  
Zhao Dong ◽  
Ke Liu ◽  
Xian-Yuan Wu
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Degree Sequence

scholarly journals How to determine if a random graph with a fixed degree sequence has a giant component

2017 ◽  
Vol 170 (1-2) ◽  
pp. 263-310 ◽  
Author(s):  
Felix Joos ◽  
Guillem Perarnau ◽  
Dieter Rautenbach ◽  
Bruce Reed
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Giant Component ◽  
Fixed Degree

scholarly journals Critical Window for Connectivity in the Configuration Model

2017 ◽  
Vol 26 (5) ◽  
pp. 660-680 ◽  
Author(s):  
LORENZO FEDERICO ◽  
REMCO VAN DER HOFSTAD
Keyword(s):  
Degree Sequence ◽  
Random Variable ◽  
Giant Component ◽  
Configuration Model ◽  
Moment Condition ◽  
Connected Graphs ◽  
Critical Window ◽  
Connectivity Probability ◽  
Asymptotic Number ◽  
Asymptotic Probability

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

scholarly journals The Critical Phase for Random Graphs with a Given Degree Sequence

2008 ◽  
Vol 17 (1) ◽  
pp. 67-86 ◽  
Author(s):  
M. KANG ◽  
T. G. SEIERSTAD
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Branching Process ◽  
Degree Sequence ◽  
Singularity Analysis ◽  
Fixed Degree ◽  
Critical Phase ◽  
Before And After

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

Ensembles with hard constraints

2017 ◽  
Author(s):  
A.C.C. Coolen ◽  
A. Annibale ◽  
E.S. Roberts
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Partition Function ◽  
Statistical Mechanics ◽  
Random Graph ◽  
Degree Sequence ◽  
Fixed Degree ◽  
Eigenvalue Spectrum ◽  
Hard Constraint ◽  
Hard Constraints

This chapter introduces random graph ensembles involving hard constraints such as setting a fixed total number of links or fixed degree sequence, including properties of the partition function. It continues on from the previous chapter’s investigation of ensembles with soft-constrained numbers of two-stars (two-step paths) and soft-constrained total number of triangles, but now combined with a hard constraint on the total number of links. This illustrates phase transitions in a mixed-constrained ensemble – which in this case is shown to be a condensation transition, where the network becomes clumped. This is investigated in detail using techniques from statistical mechanics and also looking at the averaged eigenvalue spectrum of the ensemble. These phase transition phenomena have important implications for the design of graph generation algorithms. Although hard constraints can (by force) impose required values of observables, difficult-to-reconcile constraints can lead to graphs being generated with unexpected and unphysical overall topologies.

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 $(np-1)^3n\to\infty$.

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