Critical behavior in 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.

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Parameterized clique on inhomogeneous random graphs

Discrete Applied Mathematics ◽

10.1016/j.dam.2014.10.018 ◽

2015 ◽

Vol 184 ◽

pp. 130-138 ◽

Cited By ~ 3

Author(s):

Tobias Friedrich ◽

Anton Krohmer

Keyword(s):

Random Graphs ◽

Inhomogeneous Random Graphs

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The Largest Component in Subcritical Inhomogeneous Random Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548310000180 ◽

2010 ◽

Vol 20 (1) ◽

pp. 131-154 ◽

Cited By ~ 8

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.

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Duality in inhomogeneous random graphs, and the cut metric

Random Structures and Algorithms ◽

10.1002/rsa.20348 ◽

2010 ◽

Vol 39 (3) ◽

pp. 399-411 ◽

Cited By ~ 2

Author(s):

Svante Janson ◽

Oliver Riordan

Keyword(s):

Random Graphs ◽

Inhomogeneous Random Graphs

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Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

Random Structures and Algorithms ◽

10.1002/rsa.20503 ◽

2013 ◽

Vol 43 (4) ◽

pp. 486-539 ◽

Cited By ~ 5

Author(s):

Tatyana S. Turova

Keyword(s):

Random Graphs ◽

Diffusion Approximation ◽

Inhomogeneous Random Graphs

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Geometric inhomogeneous random graphs

Theoretical Computer Science ◽

10.1016/j.tcs.2018.08.014 ◽

2019 ◽

Vol 760 ◽

pp. 35-54 ◽

Cited By ~ 10

Author(s):

Karl Bringmann ◽

Ralph Keusch ◽

Johannes Lengler

Keyword(s):

Random Graphs ◽

Inhomogeneous Random Graphs

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Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems

Probability Theory and Related Fields ◽

10.1007/s00440-021-01075-z ◽

2021 ◽

Author(s):

Nicolas Broutin ◽

Thomas Duquesne ◽

Minmin Wang

Keyword(s):

Random Graphs ◽

Metric Spaces ◽

Companion Paper ◽

Explicit Construction ◽

Convergence Condition ◽

Asymptotic Distributions ◽

Connected Components ◽

Inhomogeneous Random Graphs ◽

The Masses ◽

The Continuum

AbstractWe consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs. arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), the asymptotic homogeneous case as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or the power-law case as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

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