scholarly journals Percolation on Random Graphs with a Fixed Degree Sequence

2022 ◽  
Vol 36 (1) ◽  
pp. 1-46
Author(s):  
Nikolaos Fountoulakis ◽  
Felix Joos ◽  
Guillem Perarnau
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Fixed Degree

scholarly journals Karp–Sipser on Random Graphs with a Fixed Degree Sequence

2011 ◽  
Vol 20 (5) ◽  
pp. 721-741 ◽  
Author(s):  
TOM BOHMAN ◽  
ALAN FRIEZE
Keyword(s):  
Random Graphs ◽  
High Probability ◽  
Degree Sequence ◽  
Fixed Degree

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

scholarly journals Hamilton Cycles in Random Graphs with a Fixed Degree Sequence

2010 ◽  
Vol 24 (2) ◽  
pp. 558-569 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Michael Krivelevich
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Hamilton Cycles ◽  
Fixed Degree

SIR epidemics on random graphs with a fixed degree sequence

2012 ◽  
Vol 41 (2) ◽  
pp. 179-214 ◽  
Author(s):  
Tom Bohman ◽  
Michael Picollelli
Keyword(s):  
Random Graphs ◽  
Degree Sequence ◽  
Fixed Degree

scholarly journals On the Chromatic Number of Random Graphs with a Fixed Degree Sequence

2007 ◽  
Vol 16 (05) ◽  
pp. 733 ◽  
Author(s):  
ALAN FRIEZE ◽  
MICHAEL KRIVELEVICH ◽  
CLIFF SMYTH
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Degree Sequence ◽  
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 Random graphs with a given degree sequence

2011 ◽  
Vol 21 (4) ◽  
pp. 1400-1435 ◽  
Author(s):  
Sourav Chatterjee ◽  
Persi Diaconis ◽  
Allan Sly
Keyword(s):  
Random Graphs ◽  
Degree Sequence

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.

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

Directed weighted random graphs with an increasing bi-degree sequence

2016 ◽  
Vol 119 ◽  
pp. 235-240 ◽  
Author(s):  
Zhang Yong ◽  
Siyu Chen ◽  
Hong Qin ◽  
Ting Yan
Keyword(s):  
Random Graphs ◽  
Degree Sequence
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