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 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.

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.

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.

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.

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 On the Random Satisfiable Process

2009 ◽  
Vol 18 (5) ◽  
pp. 775-801 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV ◽  
DAN VILENCHIK
Keyword(s):  
Degree Sequence ◽  
Natural Extension ◽  
Solution Space ◽  
Polynomial Time Algorithm ◽  
Random Processes ◽  
Time Algorithm ◽  
Satisfying Assignment ◽  
Fixed Degree ◽  
Graph Properties ◽  
Cnf Formulas

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas. randomly permute all $2^k\binom{n}{k}$ possible clauses over the variables x1,. . .,xn, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if, after its addition, the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order).Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruciński and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erdős, Suen and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties have been studied, such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting.Our main contribution is as follows. For m ≥ cn, c = c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e−Ω(m/n)n of the variables take the same value in all satisfying assignments. We also describe a polynomial-time algorithm that finds w.h.p. a satisfying assignment for such formulas.

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