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.

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 Scaling Limits of Random Graphs from Subcritical Classes: Extended abstract

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Konstantinos Panagiotou ◽  
Benedikt Stufler ◽  
Kerstin Weller
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Scaling Factor ◽  
Scaling Limits ◽  
First Passage Percolation ◽  
First Passage ◽  
Outerplanar Graphs ◽  
Nous Montrons ◽  
Analytic Expressions ◽  
International Audience

International audience We study the uniform random graph $\mathsf{C}_n$ with $n$ vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph $\mathsf{C}_n / \sqrt{n}$ converges to the Brownian Continuum Random Tree $\mathcal{T}_{\mathsf{e}}$ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter $\text{D}(\mathsf{C}_n)$ and height $\text{H}(\mathsf{C}_n^\bullet)$ of the rooted random graph $\mathsf{C}_n^\bullet$. We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on $\mathsf{C}_n$, where we show the convergence to $\mathcal{T}_{\mathsf{e}}$ under an appropriate rescaling. On s’int´eresse au comportement asymptotique du graphe aleatoire $\mathsf{C}_n$ sur $n$ sommets pris uniformément d’une classe sous-critique des graphes sur n sommets. Dans cette contribution nous montrons que le graphe normalisée$\mathsf{C}_n / \sqrt{n}$ converges vers un arbre aléatoire brownien continue Te multiplie par une constante qui dépends de la classede graphes considérée. Nous calculons l’expression analytique pour cette constante dans plusieurs cas parmi la classefameuse des graphes planaire extérieure. En plus, on montre que le diamètre $\text{D}(\mathsf{C}_n)$ et la hauteur $\text{H}(\mathsf{C}_n^\bullet)$ de l’équivalent racine de $\mathsf{C}_n$ sont bornes par des bornes sous gaussiens. Notre méthode nous permettons aussi de l’étudier la percolation du premier passage sur $\mathsf{C}_n$. Nous montrons que $\mathcal{T}_{\mathsf{e}}$ sujet a une changement d’échelle appropriée

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.

A Note on a Problem by Welsh in First-Passage Percolation

1998 ◽  
Vol 7 (1) ◽  
pp. 11-15 ◽  
Author(s):  
SVEN ERICK ALM
Keyword(s):  
First Passage Times ◽  
Square Lattice ◽  
Trivial Case ◽  
First Passage Percolation ◽  
First Passage ◽  
General Proof ◽  
Mean First Passage Times ◽  
The Subject ◽  
Passage Times

Consider first-passage percolation on the square lattice. Welsh, who together with Hammersley introduced the subject in 1963, has formulated a problem about mean first-passage times, which, although seemingly simple, has not been proved in any non-trivial case. In this paper we give a general proof of Welsh's problem.

First-passage percolation on the square lattice, II

1977 ◽  
Vol 9 (02) ◽  
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.

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