Anomalous diffusion and long-range memory in the scaled voter model

Paper: arXiv 2301.08088

Authors: Rytis Kazakevičius, Aleksejus Kononovicius

Abstract

We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model, but it is driven by the scaled Brownian motion. We derive analytical expressions for the time evolution of the first and second moments of the scaled voter model. In addition, we have derived an analytical approximation of the first passage time distribution. By numerical simulation, we confirm our analytical results as well as show that the model exhibits long-range memory indicators despite being a Markov model. The proposed model has steady-state distribution consistent with the bounded fractional Brownian motion, thus we expect it to be a good substitute model for the bounded fractional Brownian motion.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents strong theoretical and mathematical derivations regarding the scaled voter model, moments, and first passage time distributions. However, empirical rigor is limited as the findings rely primarily on numerical simulations rather than backtesting on real-world market datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Model time-dependent herding in voter models & examine memory effects] --> B[Define Scaled Voter Model with power-law herding intensity]
    B --> C[Analytical Derivations: Moments & First Passage Time Distribution]
    B --> D[Numerical Simulations & Stochastic Analysis]
    C --> E[Key Finding: Exact formulas for 1st/2nd moments & FPTD approximation]
    D --> E
    E --> F[Key Outcome: Model exhibits anomalous diffusion & long-range memory]
    F --> G[Application: Serves as a Markovian alternative to bounded fractional Brownian motion]