scholarly journals A limit theorem for small cliques in inhomogeneous random graphs

2021 ◽  
Author(s):  
Jan Hladký ◽  
Christos Pelekis ◽  
Matas Šileikis
Keyword(s):  
Limit Theorem ◽  
Random Graphs ◽  
Inhomogeneous Random Graphs

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

scholarly journals Geometric inhomogeneous random graphs

2019 ◽  
Vol 760 ◽  
pp. 35-54 ◽  
Author(s):  
Karl Bringmann ◽  
Ralph Keusch ◽  
Johannes Lengler
Keyword(s):  
Random Graphs ◽  
Inhomogeneous Random Graphs

scholarly journals First Passage Percolation on Inhomogeneous Random Graphs

2015 ◽  
Vol 47 (2) ◽  
pp. 589-610 ◽  
Author(s):  
István Kolossváry ◽  
Júlia Komjáthy
Keyword(s):  
Limit Theorem ◽  
Graph Model ◽  
Exponential Weight ◽  
First Passage Percolation ◽  
First Passage ◽  
Full Generality ◽  
Random Graph Model ◽  
Minimal Weight ◽  
Inhomogeneous Random Graphs ◽  
Inhomogeneous Random Graph

In this paper we investigate first passage percolation on an inhomogeneous random graph model introduced by Bollobás et al. (2007). Each vertex in the graph has a type from a type space, and edge probabilities are independent, but depend on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal-weight path, properly normalized, follows a central limit theorem. We handle the cases where the average number of neighbors λ̃n of a vertex tends to a finite λ̃ in full generality and consider λ̃ = ∞ under mild assumptions. This paper is a generalization of the paper of Bhamidi et al. (2011), where first passage percolation is explored on the Erdős-Rényi graphs.

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