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. 

GROSS–WITTEN POINT AND DECONFINEMENT

2005 ◽  
Vol 20 (19) ◽  
pp. 4469-4474 ◽  
Author(s):  
ROBERT D. PISARSKI
Keyword(s):  
Phase Transition ◽  
Gauge Theory ◽  
Critical Point ◽  
Mean Field ◽  
Field Approximation ◽  
Mean Field Approximation ◽  
First Order

Following Aharony et al., we analyze the deconfining phase transition in a SU(∞) gauge theory in mean field approximation. The Gross–Witten model emerges as an "ultra"-critical point for deconfinement: while thermodynamically of first order, masses vanish, asymmetrically, at the transition. Potentials for N = 3 are also shown.

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (4) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

First-order liquid-liquid phase transition in compressed hydrogen and critical point

2019 ◽  
Vol 150 (20) ◽  
pp. 204114 ◽  
Author(s):  
Chunling Tian ◽  
Fusheng Liu ◽  
Hongkuan Yuan ◽  
Hong Chen ◽  
Anlong Kuan
Keyword(s):  
Phase Transition ◽  
Liquid Phase ◽  
Critical Point ◽  
Liquid Phase Transition ◽  
First Order

scholarly journals Contagions in random networks with overlapping communities

2015 ◽  
Vol 47 (04) ◽  
pp. 973-988 ◽  
Author(s):  
Emilie Coupechoux ◽  
Marc Lelarge
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Branching Processes ◽  
Graph Model ◽  
Single Individual ◽  
Overlapping Communities ◽  
Random Graph Model ◽  
The Mean

We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

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.

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