scholarly journals Speeding up non-Markovian first-passage percolation with a few extra edges

2018 ◽  
Vol 50 (3) ◽  
pp. 858-886 ◽  
Author(s):  
Alexey Medvedev ◽  
Gábor Pete
Keyword(s):  
Random Graphs ◽  
Spanning Trees ◽  
Real Life ◽  
First Passage Percolation ◽  
Power Law Distribution ◽  
First Passage ◽  
Critical Window ◽  
Expected Time ◽  
Si Model ◽  
Passage Times

Abstract One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ>t)∼t-α with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

scholarly journals First passage percolation on sparse random graphs with boundary weights

2019 ◽  
Vol 56 (2) ◽  
pp. 458-471
Author(s):  
Lasse Leskelä ◽  
Hoa Ngo
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Time Scale ◽  
Network Size ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Connectivity Structure ◽  
Random Weights ◽  
Passage Times

AbstractA large and sparse random graph with independent exponentially distributed link weights can be used to model the propagation of messages or diseases in a network with an unknown connectivity structure. In this article we study an extended setting where, in addition, the nodes of the graph are equipped with nonnegative random weights which are used to model the effect of boundary delays across paths in the network. Our main results provide approximative formulas for typical first passage times, typical flooding times, and maximum flooding times in the extended setting, over a time scale logarithmic with respect to the network size.

scholarly journals Diameters in Supercritical Random Graphs Via First Passage Percolation

2010 ◽  
Vol 19 (5-6) ◽  
pp. 729-751 ◽  
Author(s):  
JIAN DING ◽  
JEONG HAN KIM ◽  
EYAL LUBETZKY ◽  
YUVAL PERES
Keyword(s):  
Lower Bounds ◽  
High Probability ◽  
Upper And Lower Bounds ◽  
First Passage Percolation ◽  
First Passage ◽  
Supercritical Regime ◽  
Critical Window ◽  
Supercritical Phase ◽  
Structure Result ◽  
Passage Times

We study the diameter of 1, the largest component of the Erdős–Rényi random graph (n, p) in the emerging supercritical phase, i.e., for p = $\frac{1+\epsilon}n$ where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε > 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of $\frac{1000}7$.We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of 1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of 1 is w.h.p. asymptotic to $\frac23 D(\epsilon,n)$, and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to $\frac{5}9D(\epsilon,n)$.

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.

scholarly journals A random hierarchical lattice: the series-parallel graph and its properties

2004 ◽  
Vol 36 (03) ◽  
pp. 824-838 ◽  
Author(s):  
B. M. Hambly ◽  
Jonathan Jordan
Keyword(s):  
Phase Transition ◽  
Random Graphs ◽  
Effective Resistance ◽  
First Passage Percolation ◽  
Hierarchical Lattice ◽  
First Passage ◽  
In Series ◽  
Parallel Graph

We consider a sequence of random graphs constructed by a hierarchical procedure. The construction replaces existing edges by pairs of edges in series or parallel with probability p. We investigate the effective resistance across the graphs, first-passage percolation on the graphs and the Cheeger constants of the graphs as the number of edges tends to infinity. In each case we find a phase transition at

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (4) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

Weak moment conditions for time coordinates in first-passage percolation models

1980 ◽  
Vol 17 (04) ◽  
pp. 968-978 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Time Constant ◽  
First Passage Times ◽  
First Passage Percolation ◽  
First Passage ◽  
Moment Conditions ◽  
Positive Order ◽  
Finite Moment ◽  
Time Coordinate ◽  
Point To Point ◽  
Passage Times

A generalization of first-passage percolation theory proves that the fundamental convergence theorems hold provided only that the time coordinate distribution has a finite moment of a positive order. The existence of a time constant is proved by considering first-passage times between intervals of sites, rather than the usual point-to-point and point-to-line first-passage times. The basic limit theorems for the related stochastic processes follow easily by previous techniques. The time constant is evaluated as 0 when the atom at 0 of the time-coordinate distribution exceeds½.

First-passage percolation on the square lattice, II

1977 ◽  
Vol 9 (2) ◽  
pp. 283-295 ◽  
Author(s):  
John C. Wierman
Keyword(s):  
Sufficient Conditions ◽  
Initial Result ◽  
First Passage Times ◽  
Square Lattice ◽  
Maximum Height ◽  
First Passage Percolation ◽  
First Passage ◽  
Necessary And Sufficient ◽  
Passage Times

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.

scholarly journals First passage percolation on random graphs with finite mean degrees

2010 ◽  
Vol 20 (5) ◽  
pp. 1907-1965 ◽  
Author(s):  
Shankar Bhamidi ◽  
Remco van der Hofstad ◽  
Gerard Hooghiemstra
Keyword(s):  
Random Graphs ◽  
First Passage Percolation ◽  
First Passage

scholarly journals First passage percolation and a model for competing spatial growth

1998 ◽  
Vol 35 (03) ◽  
pp. 683-692 ◽  
Author(s):  
Olle Häggström ◽  
Robin Pemantle
Keyword(s):  
Spanning Tree ◽  
Particle System ◽  
Single Site ◽  
System Modelling ◽  
Positive Probability ◽  
First Passage Percolation ◽  
First Passage ◽  
Interacting Particle ◽  
State 1 ◽  
Passage Times

An interacting particle system modelling competing growth on the ℤ2 lattice is defined as follows. Each x ∈ ℤ2 is in one of the states {0,1,2}. 1's and 2's remain in their states for ever, while a 0 flips to a 1 (a 2) at a rate equal to the number of its neighbours which are in state 1 (2). This is a generalization of the well-known Richardson model. 1's and 2's may be thought of as two types of infection, and 0's as uninfected sites. We prove that if we start with a single site in state 1 and a single site in state 2, then there is positive probability for the event that both types of infection reach infinitely many sites. This result implies that the spanning tree of time-minimizing paths from the origin in first passage percolation with exponential passage times has at least two topological ends with positive probability.

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