Number of distinct sites visited by<i>N</i>random walkers

Larralde, Hernán; Trunfio, Paul; Havlin, Shlomo; Stanley, H. Eugene; Weiss, G. H.

doi:10.1103/physreva.45.7128

Cited by 96 publications

(57 citation statements)

References 21 publications

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“…Such studies were first concerned with the properties of first passage times [27][28][29][30]. The study of the mean number of distinct sites visited by ν random walkers, s ∞,ν (t), was initiated in [31] where asymptotic expressions for ν large were obtained. In [32,33] some corrections to [31] were given and subdominant contributions were evaluated.…”

Section: Introductionmentioning

confidence: 99%

Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

Turban¹

2014

J. Phys. A: Math. Theor.

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Abstract. The probability distribution of the number s of distinct sites visited up to time t by a random walk on the fully-connected lattice with N sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. Then, using generating function techniques, we compute the joint probability distribution of s and r, where r is the number of sites visited only once up to time t. Mean values, variances and covariance are deduced from the generating functions and their finitesize-scaling behaviour is studied. Introducing properly centered and scaled variables u and v for r and s and working in the scaling limit (t → ∞, N → ∞ with w = t/N fixed) the joint probability density of u and v is shown to be a bivariate Gaussian density. It follows that the fluctuations of r and s around their mean values in a finite-size system are Gaussian in the scaling limit. The same type of finite-size scaling is expected to hold on periodic lattices above the critical dimension d c = 2.

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Section: Introductionmentioning

confidence: 99%

Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

Turban¹

2014

J. Phys. A: Math. Theor.

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“…Previous researchers have noted the maximum distance from the origin to a covered node and the minimum distance to an uncovered node [34], as well as the roughening of the boundary inside the annulus determined by these two radii [31].…”

Section: Coverages -K-random Walkersmentioning

confidence: 99%

“…For example, the number of distinct sites visited by the walkers as well as the shape of the covered space by the large number of simultaneous multiple random walkers (initiated from a single node) were studied in [31][32][33][34]. Lawler [35] studied the probability of the intersection of random walk of length t in 4D.…”

Section: Introductionmentioning

confidence: 99%

Coverage maximization under resource constraints using proliferating random walks

2015

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Dissemination of information has been one of the prime needs in almost every kind of communication network. The existing algorithms for this service, try to maximize the coverage, i.e., the number of distinct nodes to which a given piece of information could be conveyed under the constraints of time and energy. However, the problem becomes challenging for unstructured and decentralized environments. Due to its simplicity and adaptability, random walk (RW) has been a very useful tool for such environments. Different variants of this technique have been studied. In this paper, we study a history-based non-uniform proliferating random strategy where new walkers are dynamically introduced in the sparse regions of the network. Apart from this, we also study the breadth-first characteristics of the random walk-based algorithms through an appropriately designed metrics.

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“…From standard results on a random walk [15], this is of order Θ(n/ log(n)) as discussed earlier in Section 7.…”

Section: Spatial Spreading and Memory Requirementsmentioning

confidence: 91%

“…Thus, this strategy will require memory to scale more slowly than the number of nodes per unit area (i.e., memory in o(n) nodes will be used to aid the query), as the Lebesgue measure of the Brownian trajectory is zero. In fact, the average number of distinct nodes that a random walk traverses over n time-steps scales as n log(n) (see [15]). …”

Section: Memory Comparison Of Sticky Search and Cachingmentioning

confidence: 99%

Asymptotics of query strategies over a sensor network

Shakkottai

Ieee Infocom 2004

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We consider the problem of a user querying for information over a sensor network, where the user does not have prior knowledge of the location of the information. We consider three information query strategies: (i) a Source-only search, where the source (user) tries to locate the destination by initiating query which propagates as a continuous time random walk (Brownian motion); (ii) a Source and Receiver Driven "Sticky" Search, where both the source and the destination send a query or an advertisement, and these leave a "sticky" trail to aid in locating the destination; and (iii) where the destination information is spatially cached (i.e., repeated over space), and the source tries to locate any one of the caches. After a random interval of time with average t, if the information is not located, the query times-out, and the search is unsuccessful.For a source-only search, we show that the probability that a query is unsuccessful decays as (log(t)) −1 . When both the source and the destination send queries or advertisements, we show that the probability that a query is unsuccessful decays as t −5/8 . Further, faster polynomial decay rates can be achieved by using a finite number of queries or advertisements. Finally, when a spatially periodic cache is employed, we show that the probability that a query is unsuccessful decays no faster than t −1 . Thus, we can match the decay rates of the source and the destination driven search with that of a spatial caching strategy by using an appropriate number of queries.The sticky search as well as caching utilize memory that is spatially distributed over the network. We show that spreading the memory over space leads to a decrease in memory requirement, while maintaining a polynomial decay in the query failure probability. In particular, we show that the memory requirement for spatial caching is larger (in an order sense) than that for sticky searches. This indicates that the appropriate strategy for querying over large sensor networks with little infrastructure support would be to use multiple queries and advertisements using the sticky search strategy. *

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Number of distinct sites visited byNrandom walkers

Cited by 96 publications

References 21 publications

Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

Coverage maximization under resource constraints using proliferating random walks

Asymptotics of query strategies over a sensor network

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