scholarly journals First Passage Percolation on Inhomogeneous Random Graphs

2015 ◽  
Vol 47 (02) ◽  
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 Bollobáset 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 λ̃nof 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 Bhamidiet al.(2011), where first passage percolation is explored on the Erdős-Rényi 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.

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 Flooding and diameter in general weighted random graphs

2020 ◽  
Vol 57 (3) ◽  
pp. 956-980
Author(s):  
Thomas Mountford ◽  
Jacques Saliba
Keyword(s):  
Asymptotic Behavior ◽  
Graph Model ◽  
Configuration Model ◽  
Tail Behavior ◽  
First Passage Percolation ◽  
Exponential Tail ◽  
First Passage ◽  
Random Graph Model ◽  
Optimal Paths ◽  
Flooding Time

AbstractIn this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.

scholarly journals Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks

2010 ◽  
Vol 42 (03) ◽  
pp. 706-738 ◽  
Author(s):  
Shankar Bhamidi ◽  
Remco van der Hofstad ◽  
Gerard Hooghiemstra
Keyword(s):  
Dirichlet Distribution ◽  
Random Network ◽  
Mean Field ◽  
Network Models ◽  
Value Theory ◽  
Configuration Model ◽  
Exponential Weight ◽  
First Passage Percolation ◽  
First Passage ◽  
Minimal Weight

We study first passage percolation (FPP) on the configuration model (CM) having power-law degrees with exponent τ ∈ [1, 2) and exponential edge weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal-weight path, both of which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via construction of infinite limiting objects describing the FPP problem in the densely connected core of the network. We consider two separate cases, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, and the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly, the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as τ > 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that information can be transferred remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der Hofstad and Hooghiemstra (2009).

scholarly journals Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij })

2012 ◽  
Vol 44 (01) ◽  
pp. 139-165
Author(s):  
Kaisheng Lin ◽  
Gesine Reinert
Keyword(s):  
Random Graph ◽  
Power Law ◽  
Graph Model ◽  
Multivariate Normal ◽  
Random Graph Model ◽  
Inhomogeneous Random Graphs ◽  
Vertex Degrees ◽  
Law Degree ◽  
Process Approximation ◽  
Inhomogeneous Random Graph

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

scholarly journals Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks

2010 ◽  
Vol 42 (3) ◽  
pp. 706-738 ◽  
Author(s):  
Shankar Bhamidi ◽  
Remco van der Hofstad ◽  
Gerard Hooghiemstra
Keyword(s):  
Dirichlet Distribution ◽  
Random Network ◽  
Mean Field ◽  
Network Models ◽  
Value Theory ◽  
Configuration Model ◽  
Exponential Weight ◽  
First Passage Percolation ◽  
First Passage ◽  
Minimal Weight

We study first passage percolation (FPP) on the configuration model (CM) having power-law degrees with exponent τ ∈ [1, 2) and exponential edge weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal-weight path, both of which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via construction of infinite limiting objects describing the FPP problem in the densely connected core of the network. We consider two separate cases, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, and the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly, the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as τ > 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that information can be transferred remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der Hofstad and Hooghiemstra (2009).

scholarly journals Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij})

2012 ◽  
Vol 44 (1) ◽  
pp. 139-165
Author(s):  
Kaisheng Lin ◽  
Gesine Reinert
Keyword(s):  
Random Graph ◽  
Power Law ◽  
Graph Model ◽  
Multivariate Normal ◽  
Random Graph Model ◽  
Inhomogeneous Random Graphs ◽  
Vertex Degrees ◽  
Law Degree ◽  
Process Approximation ◽  
Inhomogeneous Random Graph

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

First-passage percolation processes with finite height

1985 ◽  
Vol 22 (4) ◽  
pp. 766-775
Author(s):  
Norbert Herrndorf
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
First Passage Percolation ◽  
Horizontal Strip ◽  
First Passage ◽  
Time Constants ◽  
Finite Height ◽  
Special Cases ◽  
Percolation Processes ◽  
Passage Times

We consider first-passage percolation in an infinite horizontal strip of finite height. Using methods from the theory of Markov chains, we prove a central limit theorem for first-passage times, and compute the time constants for some special cases.

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