Analytical and Numerical Treatment of Continuous Ageing in the Voter Model
Abstract: The conventional voter model is modified so that an agent’s switching rate depends on the `age’ of the agent—that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. The thinning algorithm of Lewis and Shedler can be modified in order to provide an efficient simulation method. A… Show more
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 8 publications
References 64 publications
Aging properties of the voter model with long-range interactions
We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S i = ± 1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P ( r ) ∝ r − α . Employing both analytical and numerical methods, we compute the two-time correlation function G ( r ; t , s ) ( t ⩾ s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A ( t , s ) = G ( r = 0 ; t , s ) , decays algebraically for α > 1 as [ L ( t ) / L ( s ) ] − λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ = 1 / ( α − 1 ) for 1 < α ⩽ 2 . For α ⩽ 1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G ( r ; t , s ) obeys a scaling symmetry G ( r ; t , s ) = g [ r / L ( s ) ; L ( t ) / L ( s ) ] for α > 2. Similarly, for 1 < α ⩽ 2 , one has G ( r ; t , s ) = g [ r / L ( t ) ; L ( t ) / L ( s ) ] , where the length L regulating two-time correlations now differs from the coherence length as L ∝ L δ , with δ = 1 + 2 ( 2 − α ) .
Aging properties of the voter model with long-range interactions
We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, S i = ± 1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P ( r ) ∝ r − α . Employing both analytical and numerical methods, we compute the two-time correlation function G ( r ; t , s ) ( t ⩾ s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A ( t , s ) = G ( r = 0 ; t , s ) , decays algebraically for α > 1 as [ L ( t ) / L ( s ) ] − λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ = 1 / ( α − 1 ) for 1 < α ⩽ 2 . For α ⩽ 1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G ( r ; t , s ) obeys a scaling symmetry G ( r ; t , s ) = g [ r / L ( s ) ; L ( t ) / L ( s ) ] for α > 2. Similarly, for 1 < α ⩽ 2 , one has G ( r ; t , s ) = g [ r / L ( t ) ; L ( t ) / L ( s ) ] , where the length L regulating two-time correlations now differs from the coherence length as L ∝ L δ , with δ = 1 + 2 ( 2 − α ) .
A three-state language competition model including language learning and attrition
We develop a three-state agent-based language competition model that takes into account the fact that language learning and attrition are not instantaneous but occur over a finite time interval; i.e., we introduce memory in the system. We show that memory effects significantly impact the dynamics of language competition. Furthermore, we find that including heterogeneity in the linguistic skills of the agents affects the results substantially. We also explore the role of other factors, such as different levels of language learning difficulty, initial population fractions, and daily interaction rates.
Ordering kinetics with long-range interactions: interpolating between voter and Ising models
We study the ordering kinetics of a generalization of the voter model with long-range interactions, the p-voter model, in one dimension. It is defined in terms of Boolean variables S i , agents or spins, located on sites i of a lattice, each of which takes in an elementary move the state of the majority of p other agents at distances r chosen with probability P ( r ) ∝ r − α . For p = 2 the model can be exactly mapped onto the case with p = 1, which amounts to the voter model with long-range interactions decaying algebraically. For 3 ⩽ p < ∞ , instead, the dynamics falls into the universality class of the one-dimensional Ising model with long-ranged coupling constant J ( r ) = P ( r ) quenched to small finite temperatures. In the limit p → ∞ , a crossover to the (different) behavior of the long-range Ising model quenched to zero temperature is observed. Since for p > 3 a closed set of differential equations cannot be found, we employed numerical simulations to address this case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
