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Register a new userRandom walkers with extreme value memory: modelling the peak-end rule
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Rosemary J. Harris
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Motivated by the psychological literature on the peak-end rule for remembered experience, we perform an analysis within a random walk framework of a discrete choice model where agents future choices depend on the peak memory of their past experiences. In particular, we use this approach to investigate whether increased noise/disruption always leads to more switching between decisions. Here extreme value theory illuminates different classes of dynamics indicating that the long-time behaviour is dependent on the scale used for reflection; this could have implications, for example, in questionnaire design.
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Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and cover time are known. However, many recent applications such as search engines and recommender systems involve multiple random walkers on complex networks. In this work, based on numerical simulations, we show that the fraction of nodes of scale-free network not visited by $W$ random walkers in time $t$ has a stretched exponential form independent of the details of the network and number of walkers. This leads to a power-law relation between nodes not visited by $W$ walkers and by one walker within time $t$. The problem of finding the distinct nodes visited by $W$ walkers, effectively, can be reduced to that of a single walker. The robustness of the results is demonstrated by verifying them on four different real-world networks that approximately display scale-free structure.
We consider random walkers that deform the medium as they move, enabling a faster motion in regions which have been recently visited. This induces an effective attraction between walkers mediated by the medium, which can be regarded as a space metric, giving rise to a statistical mechanics toy model either for gravity, motion through deformable matter or adaptable geometry. In the strong-deformability regime, we find that diffusion is initially described by the porous medium equation, thus yielding subdiffusive behavior of an initially localized cloud of particles. Indeed, while the average width of a single cloud will sustain a $sigmasim t^{1/2}$ growth, the combined width of the whole ensemble will grow like $sigmasim t^{1/3}$ in a certain time regime. This difference can be accounted for by the strong correlations between the particles, which we explore indirectly through the fluctuations of the center of mass of the cloud and the expected value of the experienced density, defined as the average density measured by the particles themselves.
We introduce a model of interacting Random Walk, whose hopping amplitude depends on the number of walkers/particles on the link. The mesoscopic counterpart of such a microscopic dynamics is a diffusing system whose diffusivity depends on the particle density. A non-equilibrium stationary flux can be induced by suitable boundary conditions, and we show indeed that it is mesoscopically described by a Fourier equation with a density dependent diffusivity. A simple mean-field description predicts a critical diffusivity if the hopping amplitude vanishes for a certain walker density. Actually, we evidence that, even if the density equals this pseudo-critical value, the system does not present any criticality but only a dynamical slowing down. This property is confirmed by the fact that, in spite of interaction, the particle distribution at equilibrium is simply described in terms of a product of Poissonians. For mesoscopic systems with a stationary flux, a very effect of interaction among particles consists in the amplification of fluctuations, which is especially relevant close to the pseudo-critical density. This agrees with analogous results obtained for Ising models, clarifying that larger fluctuations are induced by the dynamical slowing down and not by a genuine criticality. The consistency of this amplification effect with altered coloured noise in time series is also proved.
We study the ordering statistics of 4 random walkers on the line, obtaining a much improved estimate for the long-time decay exponent of the probability that a particle leads to time $t$; $P_{rm lead}(t)sim t^{-0.91287850}$, and that a particle lags to time $t$ (never assumes the lead); $P_{rm lag}(t)sim t^{-0.30763604}$. Exponents of several other ordering statistics for $N=4$ walkers are obtained to 8 digits accuracy as well. The subtle correlations between $n$ walkers that lag {em jointly}, out of a field of $N$, are discussed: For $N=3$ there are no correlations and $P_{rm lead}(t)sim P_{rm lag}(t)^2$. In contrast, our results rule out the possibility that $P_{rm lead}(t)sim P_{rm lag}(t)^3$ for $N=4$, though the correlations in this borderline case are tiny.
We present a simple paradigm for detection of an immobile target by a space-time coupled random walker with a finite lifetime. The motion of the walker is characterized by linear displacements at a fixed speed and exponentially distributed duration, interrupted by random changes in the direction of motion and resumption of motion in the new direction with the same speed. We call these walkers mortal creepers. A mortal creeper may die at any time during its motion according to an exponential decay law characterized by a finite mean death rate $omega_m$. While still alive, the creeper has a finite mean frequency $omega$ of change of the direction of motion. In particular, we consider the efficiency of the target search process, characterized by the probability that the creeper will eventually detect the target. Analytic results confirmed by numerical results show that there is an $omega_m$-dependent optimal frequency $omega=omega_{opt}$ that maximizes the probability of eventual target detection. We work primarily in one-dimensional ($d=1$) domains and examine the role of initial conditions and of finite domain sizes. Numerical results in $d=2$ domains confirm the existence of an optimal frequency of change of direction, thereby suggesting that the observed effects are robust to changes in dimensionality. In the $d=1$ case, explicit expressions for the probability of target detection in the long time limit are given. In the case of an infinite domain, we compute the detection probability for arbitrary times and study its early- and late-time behavior. We further consider the survival probability of the target in the presence of many independent creepers beginning their motion at the same location and at the same time. We also consider a version of the standard target problem in which many creepers start at random locations at the same time.
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