Narrow log-periodic modulations in non-Markovian random walks

Abstract

What are the necessary ingredients for log-periodicity to appear in the dynamics of a random walk model? Can they be subtle enough to be overlooked? Previous studies suggest that long-range damaged memory and negative feedback together are necessary conditions for the emergence of log-periodic oscillations. The role of negative feedback would then be crucial, forcing the system to change direction. In this paper we show that small-amplitude log-periodic oscillations can emerge when the system is driven by positive feedback. Due to their very small amplitude, these oscillations can easily be mistaken for numerical finite-size effects. The models we use consist of discrete-time random walks with strong memory correlations where the decision process is taken from memory profiles based either on a binomial distribution or on a delta distribution. Anomalous superdiffusive behavior and log-periodic modulations are shown to arise in the large time limit for convenient choices of the models parameters.

Diffusion and Memory Effect in a Stochastic Processes and the Correspondence to an Information Propagation in a Social System

“…The memory effects of anomalous diffusions are numerically simulated by Chattopadhyay [18]. The recursive relationship in a random walk model [19,20]was transformed into the Fokker-Plank equation by Silva et al to describe novel phenomena caused by memory [20]. Based on the elephant random walk model, the dependence of the diffusion on the initial condition and memory parameters is investigated [5,21].…”

A Stochastic View of Comorbidity in Speech Disorders

“…To this end, we have defined the ERW model as the standard. The reasons for our school are: (a) it has well-known diffusive regimes, (b) the propagator is known, (c) has a well-known analytical solution, and (d) is a widely used model for building other models of non-Markovian random walks [10][11][12][13][14][15][16][17][18].…”

Non-equilibrium diffusion characteristics of a particle system and the correspondence to a social system

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“…The calculating results in Refs. [8,14,18] indicate that the diffusion coefficient described by the mean square displacement is of power scaling. The value of the scaling exponent equal to, greater than and less than 1 represents free diffusion, super-diffusion and sub-diffusion, respectively.…”