Modeling the mobility with memory

Choi, Jee‐Hye; Sohn, Jang Il; Goh, K.-I.; Kim, I. M.

doi:10.1209/0295-5075/99/50001

Cited by 7 publications

(14 citation statements)

References 24 publications

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“…which gives the constant K and result (10). Figure 5 (top) shows that the exact MSD obtained numerically becomes a linear function of ln t at large times, in agreement with theory.…”

Section: A Second Momentsupporting

confidence: 76%

“…A single walker may asymptotically become localized (keeping oscillating between a few sites), or diffusive, in which the range of its position X t is infinite and the origin visited infinitely often in one dimension (1D) [1,2,6]. Numerical simulations actually show that a variety of models seems to exhibit a phase transition at finite reinforcement between a localized and a diffusive regime [7][8][9][10]. Interestingly, in the diffusive regime, the same studies have presented evidence that diffusion is anomalous, namely subdiffusive.…”

Section: Introductionmentioning

confidence: 99%

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Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

Boyer

Romo-Cruz

2014

Phys. Rev. E

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Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random-walk model with long-range memory for which not only the mean-square displacement (MSD) but also the propagator can be obtained exactly in the asymptotic limit. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anticorrelated Lévy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.

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“…which gives the constant K and result (10). Figure 5 (top) shows that the exact MSD obtained numerically becomes a linear function of ln t at large times, in agreement with theory.…”

Section: A Second Momentsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

Boyer

Romo-Cruz

2014

Phys. Rev. E

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show abstract

“…In these processes, typically, a walker on a lattice moves to a nearestneighbor site with a probability that depends on the number of times this site has been visited in the past [27][28][29]. These walks must be in principle described by a hierarchy of multiple-time distribution functions, or can be studied within field theory approaches [30].…”

mentioning

confidence: 99%

Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

2014

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Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the Central Limit Theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.PACS numbers: 05.40.Fb, 89.75.Fb, 87.23.Ge The individual displacements of living organisms exhibit rich statistical features over multiple temporal and spatial scales. Due to their seemingly erratic nature, animal movements are often interpreted as random search processes and modeled as random walks [1][2][3]. In recent years, the increasing availability of data on animal [4][5][6][7] as well as human [8][9][10][11] mobility have motivated numerous models inspired from the simple random walk (RW). Let us mention, in particular, multiple scales RWs, such as Lévy walks [12,13] or intermittent RWs [4,[14][15][16], which are walks with short local movements mixed with less frequent but longer commuting displacements.Markovian RWs are the basic paradigm for modeling animal and human mobility and they provide useful insights at short temporal scales. However, empirical studies conducted over long periods of times reveal pronounced non-Markovian effects [11,17,18]. As for humans, mounting evidence shows that many animals have sophisticated cognitive abilities and use memory to move to familiar places that are not in their immediate perception range [19,20]. The use of long-term memory should strongly impact movement and it is probably at the origin of many observations which are incompatible with RWs predictions, such as, very slow diffusion, heterogeneous space use, the tendency to revisit often particular places at the expense of others, or the emergence of routines [10,11,17,18,21,22]. Non-Markovian random walks where movement steps depend on the whole path of the walker [23-25] offer a promising modeling framework in this context. But the relative lack of available analytical results in this area limits the understanding of the effects of memory on mobility patterns.Self-attracting or reinforced RWs are an important class of non-Markovian dynamics [26]. In these processes, typically, a walker on a lattice moves to a nearestneighbor site with a probability that depends on the number of times this site has been visited in the past [27][28][29]. These walks must be in principle described by a hierarchy of multiple-time distribution functions, or can be studied within field theory approaches [30]. In a slightly different conte...

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“…(11). We should remark that the human mobility pattern revealed by tracing the travel routes of bank notes 29 or the mobile phone records 30 displays deviation from random walk; The radius of gyration of individual trajectories grows logarithmically with time 30 , in contrast to the square-root scaling in the conventional random walk, and such slow diffusion is known to arise under the memory effect [31][32][33] or the spatial quenched disorder 24 . The assumption we make about the human mobility pattern is that the coarse-grained trajectories of individuals on the time scale of t (TB) = 3 years, much longer than the previous studies, show the survival probability given in Eq.…”

Section: Fatality Rate As a Function Of Hospital Density: Modelmentioning

confidence: 99%

Optimizing hospital distribution across districts to reduce tuberculosis fatalities

Lee

Kim

Son

et al. 2020

Sci Rep

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The spatial distributions of diverse facilities are often understood in terms of the optimization of the commute distance or the economic profit. Incorporating more general objective functions into such optimization framework may be useful, helping the policy decisions to meet various social and economic demands. As an example, we consider how hospitals should be distributed to minimize the total fatalities of tuberculosis (TB). The empirical data of Korea shows that the fatality rate of TB in a district decreases with the areal density of hospitals, implying their correlation and the possibility of reducing the nationwide fatalities by adjusting the hospital distribution across districts. Approximating the fatality rate by the probability of a patient not to visit a hospital in her/his residential district for the duration period of TB and evaluating the latter probability in the random-walk framework, we obtain the fatality rate as an exponential function of the hospital density with a characteristic constant related to each district’s effective lattice constant estimable empirically. This leads us to the optimal hospital distribution which finds the hospital density in a district to be a logarithmic function of the rescaled patient density. The total fatalities is reduced by 13% with this optimum. The current hospital density deviates from the optimized one in different manners from district to district, which is analyzed in the proposed model framework. The assumptions and limitations of our study are also discussed.

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Modeling the mobility with memory

Cited by 7 publications

References 24 publications

Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

Solvable random-walk model with memory and its relations with Markovian models of anomalous diffusion

Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

Optimizing hospital distribution across districts to reduce tuberculosis fatalities

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