Phase Transition in Inhomogenous Erdős-Rényi Random Graphs via Tree Counting

Sankhya A ◽  
2017 ◽  
Vol 80 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Ghurumuruhan Ganesan
Keyword(s):  
Phase Transition ◽  
Random Graphs

scholarly journals A random hierarchical lattice: the series-parallel graph and its properties

2004 ◽  
Vol 36 (03) ◽  
pp. 824-838 ◽  
Author(s):  
B. M. Hambly ◽  
Jonathan Jordan
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Effective Resistance ◽  
First Passage Percolation ◽  
Hierarchical Lattice ◽  
First Passage ◽  
In Series ◽  
Parallel Graph

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

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 Manipulative Waiters with Probabilistic Intuition

2015 ◽  
Vol 25 (6) ◽  
pp. 823-849 ◽  
Author(s):  
MAŁGORZATA BEDNARSKA-BZDȨGA ◽  
DAN HEFETZ ◽  
MICHAEL KRIVELEVICH ◽  
TOMASZ ŁUCZAK
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Complete Graph ◽  
Absolute Constant ◽  
Perfect Information ◽  
Graph Theoretic ◽  
Graph Property ◽  
Positive Integers ◽  
Monotone Graph ◽  
Transition Type

For positive integersnandqand a monotone graph property$\mathcal{A}$, we consider the two-player, perfect information game WC(n,q,$\mathcal{A}$), which is defined as follows. The game proceeds in rounds. In each round, the first player, called Waiter, offers the second player, called Client,q+ 1 edges of the complete graphKnwhich have not been offered previously. Client then chooses one of these edges which he keeps and the remainingqedges go back to Waiter. If, at the end of the game, the graph which consists of the edges chosen by Client satisfies the property$\mathcal{A}$, then Waiter is declared the winner; otherwise Client wins the game. In this paper we study such games (also known as Picker–Chooser games) for a variety of natural graph-theoretic parameters, such as the size of a largest component or the length of a longest cycle. In particular, we describe a phase transition type phenomenon which occurs when the parameterqis close tonand is reminiscent of phase transition phenomena in random graphs. Namely, we prove that ifq⩾ (1 + ϵ)n, then Client can avoid components of ordercϵ−2lnnfor some absolute constantc> 0, whereas forq⩽ (1 − ϵ)n, Waiter can force a giant, linearly sized component in Client's graph. In the second part of the paper, we prove that Waiter can force Client's graph to be pancyclic for everyq⩽cn, wherec> 0 is an appropriate constant. Note that this behaviour is in stark contrast to the threshold for pancyclicity and Hamiltonicity of random graphs.

scholarly journals Merging percolation onZdand classical random graphs: Phase transition

2009 ◽  
pp. NA-NA
Author(s):  
Tatyana S. Turova ◽  
Thomas Vallier
Keyword(s):  
Phase Transition ◽  
Random Graphs

scholarly journals Emergent Structures in Large Networks

2013 ◽  
Vol 50 (03) ◽  
pp. 883-888 ◽  
Author(s):  
David Aristoff ◽  
Charles Radin
Keyword(s):  
Phase Transition ◽  
Large Class ◽  
Random Graph ◽  
Random Graphs ◽  
Essential Feature ◽  
Exponential Random Graph Models ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Limits ◽  
Exponential Random Graph

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al. for dense random graphs.

scholarly journals On the Potts Antiferromagnet on Random Graphs

10.37236/5975 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Amin Coja-Oghlan ◽  
Nor Jaafari
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Random Graph ◽  
Random Graphs ◽  
Math Phys

Extending a prior result of Contucci et al. [Comm. Math. Phys. 2013], we determine the free energy of the Potts antiferromagnet on the Erdős–Rényi random graph at all temperatures for average degrees $d\le (2k-1)\ln k - 2 - k^{-1/2}$. In particular, we show that for this regime of $d$ there does not occur a phase transition.

scholarly journals Percolation phase transition in weight-dependent random connection models

2021 ◽  
Vol 53 (4) ◽  
pp. 1090-1114
Author(s):  
Peter Gracar ◽  
Lukas Lüchtrath ◽  
Peter Mörters
Keyword(s):  
Phase Transition ◽  
Poisson Process ◽  
Random Graphs ◽  
Power Law ◽  
Degree Distribution ◽  
Preferential Attachment ◽  
Dimensional Space ◽  
Geometric Model ◽  
Scale Free ◽  
Power Law Exponent

AbstractWe investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

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