scholarly journals Birth of a Strongly Connected Giant in an Inhomogeneous Random Digraph

2012 ◽  
Vol 49 (3) ◽  
pp. 601-611 ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Friedrich Götze ◽  
Jerzy Jaworski
Keyword(s):  
Critical Point ◽  
General Model ◽  
Branching Process ◽  
Process Approach ◽  
Giant Component ◽  
Strongly Connected ◽  
Random Digraphs

We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and on its orientation. For this model, the critical point for the emergence of a giant component is determined via a branching process approach.

scholarly journals Birth of a Strongly Connected Giant in an Inhomogeneous Random Digraph

2012 ◽  
Vol 49 (03) ◽  
pp. 601-611 ◽  
Author(s):  
Mindaugas Bloznelis ◽  
Friedrich Götze ◽  
Jerzy Jaworski
Keyword(s):  
Critical Point ◽  
General Model ◽  
Branching Process ◽  
Process Approach ◽  
Giant Component ◽  
Strongly Connected ◽  
Random Digraphs

We present and investigate a general model for inhomogeneous random digraphs with labeled vertices, where the arcs are generated independently, and the probability of inserting an arc depends on the labels of its endpoints and on its orientation. For this model, the critical point for the emergence of a giant component is determined via a branching process approach.

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

scholarly journals Susceptibility in Inhomogeneous Random Graphs

10.37236/2035 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Svante Janson ◽  
Oliver Riordan
Keyword(s):  
Phase Transitions ◽  
Critical Point ◽  
Random Graphs ◽  
General Model ◽  
Branching Process ◽  
Linear Equations ◽  
Inhomogeneous Random Graphs ◽  
Mean Size ◽  
The Mean

We study the susceptibility, i.e., the mean size of the component containing a random vertex, in a general model of inhomogeneous random graphs. This is one of the fundamental quantities associated to (percolation) phase transitions; in practice one of its main uses is that it often gives a way of determining the critical point by solving certain linear equations. Here we relate the susceptibility of suitable random graphs to a quantity associated to the corresponding branching process, and study both quantities in various natural examples.

scholarly journals The Size of the Giant Joint Component in a Binomial Random Double Graph

10.37236/8846 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Mark Jerrum ◽  
Tamás Makai
Keyword(s):  
Phase Transition ◽  
Critical Point ◽  
Random Graphs ◽  
Spanning Tree ◽  
Branching Process ◽  
Graph Model ◽  
First Order ◽  
Blue Edge ◽  
Critical Pairs

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices.  A joint component is a maximal set of vertices that supports both a red and a blue spanning tree.  We show that there are critical pairs of red and blue edge densities at which a giant joint component appears.  In contrast to the standard binomial graph model, the phase transition is first order:  the size of the largest joint component jumps from $O(1)$ vertices to $\Theta(n)$ at the critical point.  We connect this phenomenon to the properties of a certain bicoloured branching process. 

scholarly journals Branching Process Approach for 2-Sat Thresholds

2010 ◽  
Vol 47 (03) ◽  
pp. 796-810
Author(s):  
Elchanan Mossel ◽  
Arnab Sen
Keyword(s):  
Branching Process ◽  
Process Approach ◽  
Maximum Eigenvalue ◽  
Sharp Threshold

It is well known that, as n tends to ∞, the probability of satisfiability for a random 2-SAT formula on n variables, where each clause occurs independently with probability α / 2n, exhibits a sharp threshold at α = 1. We study a more general 2-SAT model in which each clause occurs independently but with probability α i / 2n, where i ∈ {0, 1, 2} is the number of positive literals in that clause. We generalize the branching process arguments used by Verhoeven (1999) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.

scholarly journals Branching Process Model applied to the 1/f Noise (Ⅱ); Computer Simulations

2021 ◽  
Author(s):  
Takayuki Kobayashi
Keyword(s):  
Computer Simulations ◽  
Process Model ◽  
Branching Process ◽  
Branching Processes ◽  
Natural Phenomenon ◽  
Causal Relationships ◽  
Strongly Connected ◽  
Particle Counts ◽  
A Chain

In order to discuss the 1⁄f problem, the statistics of branching processes of particles in a multiplicative medium are applied to generate a series of intervals of two successive particle-counts by a detector, which has a spectrum behaving like 1⁄f over seven decades of frequency. It is also discussed that the 1⁄f fluctuations being familiar around us are strongly connected with a chain of causal relationships in a natural phenomenon.

scholarly journals A branching process approach to power markets

2019 ◽  
Vol 79 ◽  
pp. 144-156 ◽  
Author(s):  
Ying Jiao ◽  
Chunhua Ma ◽  
Simone Scotti ◽  
Carlo Sgarra
Keyword(s):  
Branching Process ◽  
Process Approach ◽  
Power Markets

Giant components in three-parameter random directed graphs

1992 ◽  
Vol 24 (4) ◽  
pp. 845-857 ◽  
Author(s):  
Tomasz Łuczak ◽  
Joel E. Cohen
Keyword(s):  
Phase Transition ◽  
Food Webs ◽  
Directed Graph ◽  
Directed Graphs ◽  
Giant Component ◽  
Critical Surface ◽  
Connected Component ◽  
Strongly Connected Component ◽  
Strongly Connected ◽  
Asymptotic Probability

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertexito vertexjwherei < j, the probability of ‘down arrows' fromitojwherei ≥ j,and the probability of bidirectional arrows betweeniandj.In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

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