First passage time and first passage percolation models for analysing network resilience and effective strategies in strategic information warfare research: a brief survey and perspective

We introduce and study a novel type of first-passage percolation problem onwhere the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probabilitypand (1 —p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e.pc>p> 1 –pc.Furthermore, we show that asp↑pcorp↓ (1 –pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

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First-passage percolation on the square lattice. I

Advances in Applied Probability ◽

10.1017/s000186780004310x ◽

1977 ◽

Vol 9 (01) ◽

pp. 38-54 ◽

Cited By ~ 5

Author(s):

R. T. Smythe ◽

John C. Wierman

Keyword(s):

Time Constant ◽

First Passage Time ◽

Passage Time ◽

Square Lattice ◽

Ergodic Theorems ◽

First Passage Percolation ◽

Critical Percolation ◽

First Passage ◽

Underlying Distribution ◽

Percolation Probability

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

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Remarks on renewal theory for percolation processes

Journal of Applied Probability ◽

10.1017/s0021900200094353 ◽

1976 ◽

Vol 13 (02) ◽

pp. 290-300 ◽

Cited By ~ 7

Author(s):

R. T. Smythe

Keyword(s):

Ergodic Theorem ◽

First Passage Time ◽

Renewal Theory ◽

Passage Time ◽

Integer Lattice ◽

Two Dimensional ◽

First Passage Percolation ◽

First Passage ◽

Underlying Distribution ◽

Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L 1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

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First-passage percolation on the square lattice. I

Advances in Applied Probability ◽

10.2307/1425815 ◽

1977 ◽

Vol 9 (1) ◽

pp. 38-54 ◽

Cited By ~ 8

Author(s):

R. T. Smythe ◽

John C. Wierman

Keyword(s):

Time Constant ◽

First Passage Time ◽

Passage Time ◽

Square Lattice ◽

Ergodic Theorems ◽

First Passage Percolation ◽

Critical Percolation ◽

First Passage ◽

Underlying Distribution ◽

Percolation Probability

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

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Asymptotics for First-Passage Times on Delaunay Triangulations

Combinatorics Probability Computing ◽

10.1017/s0963548310000477 ◽

2011 ◽

Vol 20 (3) ◽

pp. 435-453 ◽

Cited By ~ 2

Author(s):

LEANDRO P. R. PIMENTEL

Keyword(s):

First Passage Time ◽

Passage Time ◽

Upper Bounds ◽

Sufficient Condition ◽

First Passage Percolation ◽

First Passage ◽

Delaunay Triangulations ◽

Negative Constant ◽

Additivity Theory ◽

Passage Times

In this paper we study planar first-passage percolation (FPP) models on random Delaunay triangulations. In [14], Vahidi-Asl and Wierman showed, using sub-additivity theory, that the rescaled first-passage time converges to a finite and non-negative constant μ. We show a sufficient condition to ensure that μ>0 and derive some upper bounds for fluctuations. Our proofs are based on percolation ideas and on the method of martingales with bounded increments.

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Remarks on renewal theory for percolation processes

Journal of Applied Probability ◽

10.2307/3212832 ◽

1976 ◽

Vol 13 (2) ◽

pp. 290-300 ◽

Cited By ~ 11

Author(s):

R. T. Smythe

Keyword(s):

Ergodic Theorem ◽

First Passage Time ◽

Renewal Theory ◽

Passage Time ◽

Integer Lattice ◽

Two Dimensional ◽

First Passage Percolation ◽

First Passage ◽

Underlying Distribution ◽

Percolation Processes

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

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The density of interfaces: a new first-passage problem

Journal of Applied Probability ◽

10.2307/3214517 ◽

1993 ◽

Vol 30 (4) ◽

pp. 851-862 ◽

Cited By ~ 2

Author(s):

L. Chayes ◽

C. Winfield

Keyword(s):

Correlation Length ◽

First Passage Time ◽

Passage Time ◽

Scaling Behavior ◽

First Passage Percolation ◽

First Passage ◽

Site Percolation ◽

Percolation Problem ◽

First Passage Problem

We introduce and study a novel type of first-passage percolation problem on where the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probability p and (1 — p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e. pc > p > 1 – pc. Furthermore, we show that as p ↑ pc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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