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 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 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.

scholarly journals First Passage Percolation on the Erdős–Rényi Random Graph

2011 ◽  
Vol 20 (5) ◽  
pp. 683-707 ◽  
Author(s):  
SHANKAR BHAMIDI ◽  
REMCO VAN DER HOFSTAD ◽  
GERARD HOOGHIEMSTRA
Keyword(s):  
Random Graph ◽  
Branching Processes ◽  
Configuration Model ◽  
Standard Normal Distribution ◽  
First Passage Percolation ◽  
First Passage ◽  
Minimal Weight ◽  
Standard Normal ◽  
Mean And Variance ◽  
Graph Distances

In this paper we explore first passage percolation (FPP) on the Erdős–Rényi random graph Gn(pn), where we assign independent random weights, having an exponential distribution with rate 1, to the edges. In the sparse regime, i.e., when npn → λ > 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to (λ log n)/(λ − 1). Furthermore, we prove that the minimal weight centred by (log n)/(λ − 1) converges in distribution.We also investigate the dense regime, where npn → ∞. We find that although the base graph is ultra-small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever the value of pn, Hn/log n → 1 in probability and, more precisely, (Hn − βn log n)/, where βn = λn/(λn − 1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn ≫ , a case that has appeared in the literature (under stronger conditions on λn) in [4, 13]. We also find lower bounds for the maximum, over all pairs of vertices, of the optimal weight and hopcount.This paper continues the investigation of FPP initiated in [4] and [5]. Compared to the setting on the configuration model studied in [5], the proofs presented here are much simpler due to a direct relation between FPP on the Erdős–Rényi random graph and thinned continuous-time branching processes.

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.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (01) ◽  
pp. 182-209
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as thenearest neighbour graph ford,K≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

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.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 182-209 ◽  
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the nearest neighbour graph for d, K ≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

Nonuniversality of weighted random graphs with infinite variance degree

2017 ◽  
Vol 54 (1) ◽  
pp. 146-164 ◽  
Author(s):  
Enrico Baroni ◽  
Remco van der Hofstad ◽  
Júlia Komjáthy
Keyword(s):  
Continuous Time ◽  
Degree Sequence ◽  
Optimal Path ◽  
Configuration Model ◽  
Infinite Variance ◽  
Finite Variance ◽  
First Passage Percolation ◽  
First Passage ◽  
Continuous Time Processes ◽  
Graph Distances

AbstractWe prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n, as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).

scholarly journals First passage percolation in the mean field limit

2021 ◽  
Author(s):  
◽  
Adrien Schertzer
Keyword(s):  
Statistical Mechanics ◽  
Mean Field ◽  
First Passage Percolation ◽  
Euclidean Spaces ◽  
Field Limit ◽  
Mean Field Limit ◽  
First Passage ◽  
The Mean

This dissertation deals with two classical problems in statistical mechanics: the first passage percolation on Euclidean spaces, FPP for short, in both directed and undirected settings.

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