scholarly journals Corrigendum: The cover time of the giant component of a random graph, Random Structures and Algorithms 32 (2008), 401-439

2009 ◽  
Vol 34 (2) ◽  
pp. 300-304 ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
Giant Component ◽  
Cover Time ◽  
Random Structures

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

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.

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.

scholarly journals The mixing time of the giant component of a random graph

2014 ◽  
Vol 45 (3) ◽  
pp. 383-407 ◽  
Author(s):  
Itai Benjamini ◽  
Gady Kozma ◽  
Nicholas Wormald
Keyword(s):  
Random Graph ◽  
Mixing Time ◽  
Giant Component

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 power of two choices for random walks

2021 ◽  
pp. 1-28
Author(s):  
Agelos Georgakopoulos ◽  
John Haslegrave ◽  
Thomas Sauerwald ◽  
John Sylvester
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Graph ◽  
Optimal Strategy ◽  
Cover Time ◽  
Bounded Degree ◽  
Graph Classes ◽  
Algorithmic Question ◽  
Cover Times ◽  
Bounded Degree Trees

Abstract We apply the power-of-two-choices paradigm to a random walk on a graph: rather than moving to a uniform random neighbour at each step, a controller is allowed to choose from two independent uniform random neighbours. We prove that this allows the controller to significantly accelerate the hitting and cover times in several natural graph classes. In particular, we show that the cover time becomes linear in the number n of vertices on discrete tori and bounded degree trees, of order $${\mathcal O}(n\log \log n)$$ on bounded degree expanders, and of order $${\mathcal O}(n{(\log \log n)^2})$$ on the Erdős–Rényi random graph in a certain sparsely connected regime. We also consider the algorithmic question of computing an optimal strategy and prove a dichotomy in efficiency between computing strategies for hitting and cover times.

scholarly journals Cover time of a random graph with a degree sequence II: Allowing vertices of degree two

2014 ◽  
Vol 45 (4) ◽  
pp. 627-674
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Eyal Lubetzky
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Cover Time

Renewal processes, population dynamics, and unimodular trees

2019 ◽  
Vol 56 (2) ◽  
pp. 339-357
Author(s):  
François Baccelli ◽  
Antonio Sodre
Keyword(s):  
Population Dynamics ◽  
Steady State ◽  
Positive Integer ◽  
Finite Number ◽  
Random Graph ◽  
Infinite Number ◽  
Point Processes ◽  
Renewal Processes ◽  
Directed Tree ◽  
Random Structures

AbstractBased on a simple object, an i.i.d. sequence of positive integer-valued random variables {an}n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + an. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {an}n∊ℤ. Second, the directed random graph Tf on ℤ generated by the random map f(n) = n + an. Assuming only that E [a0] < ∞ and P [a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, Tf is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, Tf splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.

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