scholarly journals A phase transition regarding the evolution of bootstrap processes in inhomogeneous random graphs

2018 ◽  
Vol 28 (2) ◽  
pp. 990-1051
Author(s):  
Nikolaos Fountoulakis ◽  
Mihyun Kang ◽  
Christoph Koch ◽  
Tamás Makai
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Inhomogeneous Random Graphs

scholarly journals Phase transition in random distance graphs on the torus

2017 ◽  
Vol 54 (4) ◽  
pp. 1278-1294 ◽  
Author(s):  
Fioralba Ajazi ◽  
George M. Napolitano ◽  
Tatyana Turova
Keyword(s):  
Phase Transition ◽  
General Theory ◽  
Random Graphs ◽  
Connected Component ◽  
Subcritical Case ◽  
Distance Graphs ◽  
Inhomogeneous Random Graphs ◽  
Supercritical Phase

Abstract In this paper we consider random distance graphs motivated by applications in neurobiology. These models can be viewed as examples of inhomogeneous random graphs, notably outside of the so-called rank-1 case. Treating these models in the context of the general theory of inhomogeneous graphs helps us to derive the asymptotics for the size of the largest connected component. In particular, we show that certain random distance graphs behave exactly as the classical Erdős–Rényi model, not only in the supercritical phase (as already known) but in the subcritical case as well.

scholarly journals The phase transition in inhomogeneous random graphs

2007 ◽  
Vol 31 (1) ◽  
pp. 3-122 ◽  
Author(s):  
Béla Bollobás ◽  
Svante Janson ◽  
Oliver Riordan
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Inhomogeneous 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 Big Jobs Arrive Early: From Critical Queues to Random Graphs

2020 ◽  
Vol 10 (4) ◽  
pp. 310-334
Author(s):  
Gianmarco Bet ◽  
Remco van der Hofstad ◽  
Johan S. H. van Leeuwaarden
Keyword(s):  
Head Start ◽  
Random Graphs ◽  
Queue Length ◽  
Busy Period ◽  
Asymptotic Result ◽  
Service Requirement ◽  
Exploration Process ◽  
Asymptotic Regime ◽  
Long Period ◽  
Inhomogeneous Random Graphs

We consider a queue to which only a finite pool of n customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement S arrives to the queue after an exponentially distributed time with mean S-α for some [Formula: see text]; therefore, larger service requirements trigger customers to join earlier. This finite-pool queue interpolates between two previously studied cases: α = 0 gives the so-called [Formula: see text] queue and α = 1 is closely related to the exploration process for inhomogeneous random graphs. We consider the asymptotic regime in which the pool size n grows to infinity and establish that the scaled queue-length process converges to a diffusion process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of activity. We also describe how this first busy period of the queue gives rise to a critically connected random forest.

scholarly journals Parameterized clique on inhomogeneous random graphs

2015 ◽  
Vol 184 ◽  
pp. 130-138 ◽  
Author(s):  
Tobias Friedrich ◽  
Anton Krohmer
Keyword(s):  
Random Graphs ◽  
Inhomogeneous Random Graphs

The Largest Component in Subcritical Inhomogeneous Random Graphs

2010 ◽  
Vol 20 (1) ◽  
pp. 131-154 ◽  
Author(s):  
TATYANA S. TUROVA
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Graph Model ◽  
Exact Formula ◽  
Connected Component ◽  
Subcritical Case ◽  
Random Graph Model ◽  
Asymptotic Size ◽  
Inhomogeneous Random Graphs ◽  
Inhomogeneous Random Graph

We study the ‘rank 1 case’ of the inhomogeneous random graph model. In the subcritical case we derive an exact formula for the asymptotic size of the largest connected component scaled to log n. This result complements the corresponding known result in the supercritical case. We provide some examples of applications of the derived formula.

scholarly journals Duality in inhomogeneous random graphs, and the cut metric

2010 ◽  
Vol 39 (3) ◽  
pp. 399-411 ◽  
Author(s):  
Svante Janson ◽  
Oliver Riordan
Keyword(s):  
Random Graphs ◽  
Inhomogeneous Random Graphs
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