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 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 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 Competing first passage percolation on random graphs with finite variance degrees

2019 ◽  
Vol 55 (3) ◽  
pp. 545-559
Author(s):  
Daniel Ahlberg ◽  
Maria Deijfen ◽  
Svante Janson
Keyword(s):  
Random Graphs ◽  
Finite Variance ◽  
First Passage Percolation ◽  
First Passage

scholarly journals Heavy-tailed branching random walks on multidimensional lattices. A moment approach

2020 ◽  
pp. 1-22
Author(s):  
Anastasiya Rytova ◽  
Elena Yarovaya
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Lattice Point ◽  
Positive Eigenvalue ◽  
Infinite Variance ◽  
Finite Variance ◽  
Branching Random Walks ◽  
Heavy Tailed ◽  
Number Of Particles

We study a continuous-time branching random walk (BRW) on the lattice ℤ d , d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point β c  > 0 is found for every d ≥ 1. In particular, if β > β c the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > β c called supercritical. Classification of the BRW treated as subcritical (β < β c ) or critical (β = β c ) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤ d and of the particle population on ℤ d according to the ratio d/α.

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 &gt; 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).

Asymptotic properties of the stationary measure of a Markov branching process

1973 ◽  
Vol 10 (02) ◽  
pp. 447-450 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Asymptotic Properties ◽  
Stationary Measure ◽  
Infinite Variance ◽  
Finite Variance ◽  
Renewal Theorem ◽  
Markov Branching Process

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.

Asymptotic properties of the stationary measure of a Markov branching process

1973 ◽  
Vol 10 (2) ◽  
pp. 447-450 ◽  
Author(s):  
Y. S. Yang
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Critical Case ◽  
Asymptotic Properties ◽  
Stationary Measure ◽  
Infinite Variance ◽  
Finite Variance ◽  
Renewal Theorem ◽  
Markov Branching Process

The asymptotic properties of the unique stationary measure of a Markov branching process will be given. In the critical case with finite variance, the result can be deduced from a result for discrete time processes of Kesten, Ney and Spitzer (1966) where the proof makes use of a stronger assumption than the finiteness of variance. For the continuous time case where the stationary measure has an explicit form, we can use the discrete renewal theorem which takes care of the infinite variance case as well.

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).

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