Scaling properties of paths on graphs

Edwards, Roderick; Foxall, Eric; Perkins, Theodore J.

doi:10.13001/1081-3810.1569

Cited by 6 publications

(24 citation statements)

References 11 publications

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“…Lerner ponential order of the asymptotics). Therefore, one can consider results obtained in this paper as a strengthening of results of [9], where one has proved weak power and weak subexponential asymptotics. Note that earlier in [4] for the subexponential case, we considered only necessary and sufficient conditions of the exponential decreasing order (D = 1).…”

mentioning

confidence: 56%

“…Having completed the main part of this paper (see [4]), we became aware of the paper [9] , where under the same conditions one proves the existence of limits ln p(t)/ ln t as t → ∞ (which coincides with our case of the power order of the asymptotics, as well as with the case of the power asymptotics, where correction data change is slower) and the limit ln p(t)/t 1/D , where D ∈ N (which coincides with our case of the subex-ELA 536 V.V. Bochkarev and E.Yu.…”

mentioning

confidence: 99%

“…Note that earlier in [4] for the subexponential case, we considered only necessary and sufficient conditions of the exponential decreasing order (D = 1). We have proved the subexponential order in a general case only after getting acquainted with results of [9]. We calculate constants in the subexponent by explicit formulas, while the corresponding constants in [9] are calculated by recurrent formulas (which can be easily reduced to explicit ones).…”

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confidence: 99%

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Strong power and subexponential laws for an ordered list of trajectories of a Markov chain

Bochkarev¹,

Lerner²

2014

ELA

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Consider a homogeneous Markov chain with discrete time and with a finite set of states E 0 ,. .. , En such that the state E 0 is absorbing and states E 1 ,. .. , En are nonrecurrent. The frequencies of trajectories in this chain are studied in this paper, i.e., "words" composed of symbols E 1 ,. .. , En ending with the "space" E 0. Order the words according to their probabilities; denote by p(t) the probability of the t th word in this list. As was proved recently, in the case of an infinite list of words, in the dependence of the topology of the graph of the Markov chain, there exists either the limit ln p(t)/ ln t as t → ∞ or that of ln p(t)/t 1/D , where D ∈ N (weak power and subexponential laws). As appeared, in the latter case the decreasing order of the function p(t) is always subexponential (the strong subexponential law). In the first case, this paper describes necessary and sufficient conditions of the power order (the strong power law). These conditions are fulfilled, in particular, if the graph of the Markov chain that corresponds to states E 1 ,. .. , En is strongly connected.

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confidence: 56%

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confidence: 99%

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confidence: 99%

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Strong power and subexponential laws for an ordered list of trajectories of a Markov chain

Bochkarev¹,

Lerner²

2014

ELA

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“…These results were obtained independently in [12,14] and later refined in [13]. Namely, as appeared, the first alternative means the subexponential order of the asymptotics; that is, in this case ∃ 1 , 2 : 0 < 1 < 2 , such that…”

Section: Statement Of the Main Theorem And Its Connection Withmentioning

confidence: 92%

“…In this case, the asymptotic behavior of ( ) does not necessarily have a power order. Namely, in this case one of the two alternatives takes place [12,13]. The first variant is that there exists the limit…”

Section: Statement Of the Main Theorem And Its Connection Withmentioning

confidence: 99%

Calculation of Precise Constants in a Probability Model of Zipf’s Law Generation and Asymptotics of Sums of Multinomial Coefficients

Bochkarev

Lerner

2017

International Journal of Mathematics and Mathematical Sciences

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Let 0 , 1 , . . . , be a full set of outcomes (symbols) and let positive , = 0, . . . , , be their probabilities (∑ =0 = 1). Let us treat 0 as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the nonincreasing order of their probabilities. Let ( ) be the probability of the th word in this list. We prove that if at least one of the ratios log / log , , ∈ {1, . . . , }, is irrational, then the limit lim →∞ ( )/ −1/ exists and differs from zero; here is the root of the equation ∑ =1 = 1. The limit constant can be expressed (rather easily) in terms of the entropy of the distribution ( 1 , . . . , ).

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A scaling law for random walks on networks

et al. 2014

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The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.

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Scaling properties of paths on graphs

Cited by 6 publications

References 11 publications

Strong power and subexponential laws for an ordered list of trajectories of a Markov chain

Strong power and subexponential laws for an ordered list of trajectories of a Markov chain

Calculation of Precise Constants in a Probability Model of Zipf’s Law Generation and Asymptotics of Sums of Multinomial Coefficients

A scaling law for random walks on networks

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