1
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Corberi F, Dello Russo S, Smaldone L. Coarsening and metastability of the long-range voter model in three dimensions. Phys Rev E 2024; 110:024143. [PMID: 39295067 DOI: 10.1103/physreve.110.024143] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2024] [Accepted: 08/12/2024] [Indexed: 09/21/2024]
Abstract
We study analytically the ordering kinetics and the final metastable states in the three-dimensional long-range voter model where N agents described by a Boolean spin variable S_{i} can be found in two states (or opinion) ±1. The kinetics is such that each agent copies the opinion of another at distance r chosen with probability P(r)∝r^{-α} (α>0). In the thermodynamic limit N→∞ the system approaches a correlated metastable state without consensus, namely without full spin alignment. In such states the equal-time correlation function C(r)=〈S_{i}S_{j}〉 (where r is the i-j distance) decreases algebraically in a slow, nonintegrable way. Specifically, we find C(r)∼r^{-1}, or C(r)∼r^{-(6-α)}, or C(r)∼r^{-α} for α>5, 3<α≤5, and 0≤α≤3, respectively. In a finite system metastability is escaped after a time of order N and full ordering is eventually achieved. The dynamics leading to metastability is of the coarsening type, with an ever-increasing correlation length L(t) (for N→∞). We find L(t)∼t^{1/2} for α>5, L(t)∼t^{5/2α} for 4<α≤5, and L(t)∼t^{5/8} for 3≤α≤4. For 0≤α<3 there is not macroscopic coarsening because stationarity is reached in a microscopic time. Such results allow us to conjecture the behavior of the model for generic spatial dimension.
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2
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Latoski LCF, Dantas WG, Arenzon JJ. Opinion inertia and coarsening in the persistent voter model. Phys Rev E 2024; 109:054115. [PMID: 38907438 DOI: 10.1103/physreve.109.054115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2024] [Accepted: 04/16/2024] [Indexed: 06/24/2024]
Abstract
We consider the persistent voter model (PVM), a variant of the voter model (VM) that includes transient, dynamically induced zealots. Due to peer reinforcement, the internal confidence η_{i} of a normal voter increases in steps of size Δη. Once it surpasses a given threshold, it becomes a zealot. Its opinion remains frozen until enough interactions with the opposing opinion occur, resetting its confidence. No longer a zealot, the regular voter may change opinion once again. This mechanism of opinion inertia, though simplified, is responsible for an effective surface tension, and the PVM exhibits a crossover from a fluctuation-driven dynamics, as in the VM, to a curvature-driven one, akin to the Ising model at low temperature. The average time τ to attain consensus is nonmonotonic with respect to Δη and reaches a minimum at Δη_{min}. In this paper we elucidate the mechanisms that accelerate the system towards consensus close to Δη_{min}. Near the crossover at Δη_{min}, the intermediate region around the domains where the regular voters accumulate (the active region, AR) is large. The surface tension, albeit small, is sufficient to maintain the shape and reduce the domain fragmentation. The large size of the AR in the region of Δη_{min} has two important effects that accelerate the dynamics. First, it dislodges the zealots within the bulk of the domains. Secondly, it maximally suppresses the formation of slowly evolving stripes typical in Ising-like models. This suggests the importance of understanding the role of the AR, where opinion changes are facilitated, and the interplay between regular voters and zealots in disrupting polarized states.
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Affiliation(s)
- Luís Carlos F Latoski
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970, Porto Alegre - RS, Brazil
| | - W G Dantas
- Departamento de Ciências Exatas, EEIMVR, Universidade Federal Fluminense, CEP 27255-125, Volta Redonda - RJ, Brazil
| | - Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970, Porto Alegre - RS, Brazil
- Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, 22290-180 Rio de Janeiro - RJ, Brazil
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3
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Corberi F, Smaldone L. Ordering kinetics of the two-dimensional voter model with long-range interactions. Phys Rev E 2024; 109:034133. [PMID: 38632821 DOI: 10.1103/physreve.109.034133] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2023] [Accepted: 03/04/2024] [Indexed: 04/19/2024]
Abstract
We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance r with probability P(r)∝r^{-α}. The model is characterized by different regimes, as α is varied. For α>4, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as L(t)∝sqrt[t], until consensus is reached in a time of the order of NlnN, with N being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as ρ(t)∝1/lnt. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For 0<α≤4, standard scaling is reinstated and the correlation length increases algebraically as L(t)∝t^{1/z}, with 1/z=2/α for 3<α<4 and 1/z=2/3 for 0<α<3. In addition, for α≤3, L(t) depends on N at any time t>0. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the N→∞ limit. In finite systems, consensus is reached in a time of the order of N for any α<4.
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Affiliation(s)
- Federico Corberi
- Dipartimento di Fisica, Università di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy and INFN Sezione di Napoli, Gruppo Collegato di Salerno, Italy
| | - Luca Smaldone
- Dipartimento di Fisica, Università di Salerno, Via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy and INFN Sezione di Napoli, Gruppo Collegato di Salerno, Italy
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4
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Leonidov AV, Vasilyev SB, Vasilyeva EE. Interconnected Evolution of Epidemic and Public Vaccination Opinion. JETP LETTERS 2022; 117:83-89. [PMID: 36589319 PMCID: PMC9791622 DOI: 10.1134/s0021364022602172] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Revised: 11/07/2022] [Accepted: 11/09/2022] [Indexed: 06/17/2023]
Abstract
A new model of interconnected coevolving SIRS epidemic and public vaccination opinion pattern is presented. The underlying two-layer network contains strata corresponding to physical interactions in real space and social communications. The layer corresponding to physical interactions is constructed based on data on a real network representing communications between high school students. The evolution of people vaccination attitude is described using an Ising-type model. The model describes a non-trivial dependence of resulting epidemic dynamics on (1) noise amplitude, (2) initial opinion pattern and (3) influence of external information.
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Affiliation(s)
- A. V. Leonidov
- Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
| | - S. B. Vasilyev
- Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
| | - E. E. Vasilyeva
- Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia
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5
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Latoski LCF, Dantas WG, Arenzon JJ. Curvature-driven growth and interfacial noise in the voter model with self-induced zealots. Phys Rev E 2022; 106:014121. [PMID: 35974624 DOI: 10.1103/physreve.106.014121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 06/27/2022] [Indexed: 06/15/2023]
Abstract
We introduce a variant of the voter model in which agents may have different degrees of confidence in their opinions. Those with low confidence are normal voters whose state can change upon a single contact with a different neighboring opinion. However, confidence increases with opinion reinforcement, and above a certain threshold, these agents become zealots, irreducible agents who do not change their opinion. We show that both strategies, normal voters and zealots, may coexist (in the thermodynamical limit), leading to competition between two different kinetic mechanisms: curvature-driven growth and interfacial noise. The kinetically constrained zealots are formed well inside the clusters, away from the different opinions at the surfaces that help limit their confidence. Normal voters concentrate in a region around the interfaces, and their number, which is related to the distance between the surface and the zealotry bulk, depends on the rate at which the confidence changes. Despite this interface being rough and fragmented, typical of the voter model, the presence of zealots in the bulk of these domains induces a curvature-driven dynamics, similar to the low temperature coarsening behavior of the nonconserved Ising model after a temperature quench.
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Affiliation(s)
- Luís Carlos F Latoski
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
| | - W G Dantas
- Departamento de Ciências Exatas, EEIMVR, Universidade Federal Fluminense, CEP 27255-125, Volta Redonda, Rio de Janeiro, Brazil
| | - Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970, Porto Alegre, Rio Grande do Sul, Brazil
- Instituto Nacional de Ciência e Tecnologia-Sistemas Complexos, Rio de Janeiro, 22290-180, Rio de Janeiro, Brazil
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6
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Chen H, Yong EH. How zealots affect the energy cost for controlling complex social networks. CHAOS (WOODBURY, N.Y.) 2022; 32:063116. [PMID: 35778141 DOI: 10.1063/5.0085222] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2022] [Accepted: 05/16/2022] [Indexed: 06/15/2023]
Abstract
The controllability of complex networks may be applicable for understanding how to control a complex social network, where members share their opinions and influence one another. Previous works in this area have focused on controllability, energy cost, or optimization under the assumption that all nodes are compliant, passing on information neutrally without any preferences. However, the assumption on nodal neutrality should be reassessed, given that in networked social systems, some people may hold fast to their personal beliefs. By introducing some stubborn agents, or zealots, who hold steadfast to their beliefs and seek to influence others, the control energy is computed and compared against those without zealots. It was found that the presence of zealots alters the energy cost at a quadratic rate with respect to their own fixed beliefs. However, whether or not the zealots' presence increases or decreases the energy cost is affected by the interplay between different parameters such as the zealots' beliefs, number of drivers, final control time regimes, network effects, network dynamics, and number and configurations of neutral nodes influenced by the zealots. For example, when a network dynamics is linear but does not have conformity behavior, it could be possible for a contrarian zealot to assist in reducing control energy. With conformity behavior, a contrarian zealot always negatively affects network control by increasing energy cost. The results of this paper suggest caution when modeling real networked social systems with the controllability of networked linear dynamics since the system dynamical behavior is sensitive to parameter change.
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Affiliation(s)
- Hong Chen
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Ee Hou Yong
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
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7
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Franco GD, Marquitti FMD, Fernandes LD, Braha D, de Aguiar MAM. Shannon information criterion for low-high diversity transition in Moran and voter models. Phys Rev E 2021; 104:024315. [PMID: 34525569 DOI: 10.1103/physreve.104.024315] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/27/2021] [Accepted: 08/11/2021] [Indexed: 11/07/2022]
Abstract
Mutation and drift play opposite roles in genetics. While mutation creates diversity, drift can cause gene variants to disappear, especially when they are rare. In the absence of natural selection and migration, the balance between the drift and mutation in a well-mixed population defines its diversity. The Moran model captures the effects of these two evolutionary forces and has a counterpart in social dynamics, known as the voter model with external opinion influencers. Two extreme outcomes of the voter model dynamics are consensus and coexistence of opinions, which correspond to low and high diversity in the Moran model. Here we use a Shannon's information-theoretic approach to characterize the smooth transition between the states of consensus and coexistence of opinions in the voter model. Mapping the Moran into the voter model, we extend the results to the mutation-drift balance and characterize the transition between low and high diversity in finite populations. Describing the population as a network of connected individuals, we show that the transition between the two regimes depends on the network topology of the population and on the possible asymmetries in the mutation rates.
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Affiliation(s)
- Gabriella Dantas Franco
- Universidade Estadual de Campinas, Departamento de Física da Matéria Condensada, Instituto de Física "Gleb Wataghin", 13083-970 Campinas, São Paulo, Brazil
| | - Flavia Maria Darcie Marquitti
- Universidade Estadual de Campinas, Departamento de Física da Matéria Condensada, Instituto de Física "Gleb Wataghin", 13083-970 Campinas, São Paulo, Brazil.,Universidade Estadual de Campinas, Instituto de Biologia, 13083-862 Campinas, São Paulo, Brazil
| | - Lucas D Fernandes
- Department of Life Sciences, Imperial College London - Silwood Park, SL5 7PY Ascot-Berkshire, United Kingdom
| | - Dan Braha
- New England Complex Systems Institute, Cambridge, Massachusetts 02139, USA.,University of Massachusetts, Dartmouth, Massachusetts 02747, USA
| | - Marcus Aloizio Martinez de Aguiar
- Universidade Estadual de Campinas, Departamento de Física da Matéria Condensada, Instituto de Física "Gleb Wataghin", 13083-970 Campinas, São Paulo, Brazil
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8
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Bhat D, Redner S. Nonuniversal opinion dynamics driven by opposing external influences. Phys Rev E 2019; 100:050301. [PMID: 31869908 DOI: 10.1103/physreve.100.050301] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2019] [Indexed: 11/07/2022]
Abstract
We study the opinion dynamics of a generalized voter model in which N voters are additionally influenced by two opposing news sources whose effect is to promote political polarization. As the influence of these news sources is increased, the mean time to reach consensus scales nonuniversally as N^{α}. The parameter α quantifies the influence of the news sources and increases without bound as the news sources become increasingly influential. The time to reach a politically polarized state, in which roughly equal fractions of the populations are in each opinion state, is generally short, and the steady-state opinion distribution exhibits a transition from near consensus to a politically polarized state as a function of α.
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Affiliation(s)
- Deepak Bhat
- Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501, USA
| | - S Redner
- Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, New Mexico 87501, USA
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9
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Gastner MT, Takács K, Gulyás M, Szvetelszky Z, Oborny B. The impact of hypocrisy on opinion formation: A dynamic model. PLoS One 2019; 14:e0218729. [PMID: 31242270 PMCID: PMC6594623 DOI: 10.1371/journal.pone.0218729] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2018] [Accepted: 06/07/2019] [Indexed: 11/23/2022] Open
Abstract
Humans have a demonstrated tendency to copy or imitate the behavior and attitude of others and actively influence each other's opinions. In plenty of empirical contexts, publicly revealed opinions are not necessarily in line with internal opinions, causing complex social influence dynamics. We study to what extent hypocrisy is sustained during opinion formation and how hidden opinions change the convergence to consensus in a group. We build and analyze a modified version of the voter model with hypocrisy in a complete graph with a neutral competition between two alternatives. We compare the process from various initial conditions, varying the proportions between the two opinions in the external (revealed) and internal (hidden) layer. According to our results, hypocrisy always prolongs the time needed for reaching a consensus. In a complete graph, this time span increases linearly with group size. We find that the group-level opinion emerges in two steps: (1) a fast and directional process, during which the number of the two kinds of hypocrites equalizes; and (2) a slower, random drift of opinions. During stage (2), the ratio of opinions in the external layer is approximately equal to the ratio in the internal layer; that is, the hidden opinions do not differ significantly from the revealed ones at the group level. We furthermore find that the initial abundances of opinions, but not the initial prevalence of hypocrisy, predicts the mean consensus time and determines the opinions' probabilities of winning. These insights highlight the unimportance of hypocrisy in consensus formation under neutral conditions. Our results have important societal implications in relation to hidden voter preferences in polls and improve our understanding of opinion formation in a more realistic setting than that of conventional voter models.
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Affiliation(s)
- Michael T. Gastner
- Division of Science, Yale-NUS College, Singapore, Singapore
- MTA TK “Lendület” Research Center for Educational and Network Studies (RECENS), Hungarian Academy of Sciences, Budapest, Hungary
| | - Károly Takács
- MTA TK “Lendület” Research Center for Educational and Network Studies (RECENS), Hungarian Academy of Sciences, Budapest, Hungary
- The Institute for Analytical Sociology (IAS), Linköping University, Norrköping, Sweden
| | - Máté Gulyás
- MTA TK “Lendület” Research Center for Educational and Network Studies (RECENS), Hungarian Academy of Sciences, Budapest, Hungary
- Department of Plant Taxonomy, Ecology and Theoretical Biology, Biological Institute, Loránd Eötvös University (ELTE), Budapest, Hungary
| | - Zsuzsanna Szvetelszky
- MTA TK “Lendület” Research Center for Educational and Network Studies (RECENS), Hungarian Academy of Sciences, Budapest, Hungary
| | - Beáta Oborny
- Department of Plant Taxonomy, Ecology and Theoretical Biology, Biological Institute, Loránd Eötvös University (ELTE), Budapest, Hungary
- GINOP Sustainable Ecosystems Group, Centre for Ecological Research, Hungarian Academy of Sciences, Tihany, Hungary
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10
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Brede M, Restocchi V, Stein S. Resisting Influence: How the Strength of Predispositions to Resist Control Can Change Strategies for Optimal Opinion Control in the Voter Model. Front Robot AI 2018; 5:34. [PMID: 33500920 PMCID: PMC7805989 DOI: 10.3389/frobt.2018.00034] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2018] [Accepted: 03/14/2018] [Indexed: 11/22/2022] Open
Abstract
In this paper, we investigate influence maximization, or optimal opinion control, in a modified version of the two-state voter dynamics in which a native state and a controlled or influenced state are accounted for. We include agent predispositions to resist influence in the form of a probability q with which agents spontaneously switch back to the native state when in the controlled state. We argue that in contrast to the original voter model, optimal control in this setting depends on q: For low strength of predispositions q, optimal control should focus on hub nodes, but for large q, optimal control can be achieved by focusing on the lowest degree nodes. We investigate this transition between hub and low-degree node control for heterogeneous undirected networks and give analytical and numerical arguments for the existence of two control regimes.
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Affiliation(s)
- Markus Brede
- Agents, Interactions, and Complexity Group, ECS, University of Southampton, Southampton, United Kingdom.,Institute of Life Sciences, University of Southampton, Southampton, United Kingdom
| | - Valerio Restocchi
- Agents, Interactions, and Complexity Group, ECS, University of Southampton, Southampton, United Kingdom.,Southampton Business School, University of Southampton, Southampton, United Kingdom
| | - Sebastian Stein
- Agents, Interactions, and Complexity Group, ECS, University of Southampton, Southampton, United Kingdom
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11
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Mellor A, Mobilia M, Zia RKP. Heterogeneous out-of-equilibrium nonlinear q-voter model with zealotry. Phys Rev E 2017; 95:012104. [PMID: 28208330 DOI: 10.1103/physreve.95.012104] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2016] [Indexed: 06/06/2023]
Abstract
We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in Mellor et al. [Europhys. Lett. 113, 48001 (2016)EULEEJ0295-507510.1209/0295-5075/113/48001]. In this model, each individual supports one of two parties and is either a susceptible voter of type q_{1} or q_{2}, or is an inflexible zealot. At each time step, a q_{i}-susceptible voter (i=1,2) consults a group of q_{i} neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever q_{1}≠q_{2} and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the nonequilibrium stationary state of the system in terms of its probability distribution, nonvanishing currents, and unequal-time two-point correlation functions. We also study the switching time properties of the model by exploiting an approximate mapping onto the model of Mobilia [Phys. Rev. E 92, 012803 (2015)PLEEE81539-375510.1103/PhysRevE.92.012803] that satisfies the detailed balance, and we outline some properties of the model near criticality.
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Affiliation(s)
- Andrew Mellor
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
| | - R K P Zia
- Center for Soft Matter and Biological Physics, Department of Physics, Virginia Polytechnic Institute & State University, Blacksburg, Virginia 24061, USA
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12
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Rodriguez N, Bollen J, Ahn YY. Collective Dynamics of Belief Evolution under Cognitive Coherence and Social Conformity. PLoS One 2016; 11:e0165910. [PMID: 27812210 PMCID: PMC5094740 DOI: 10.1371/journal.pone.0165910] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2016] [Accepted: 10/12/2016] [Indexed: 12/01/2022] Open
Abstract
Human history has been marked by social instability and conflict, often driven by the irreconcilability of opposing sets of beliefs, ideologies, and religious dogmas. The dynamics of belief systems has been studied mainly from two distinct perspectives, namely how cognitive biases lead to individual belief rigidity and how social influence leads to social conformity. Here we propose a unifying framework that connects cognitive and social forces together in order to study the dynamics of societal belief evolution. Each individual is endowed with a network of interacting beliefs that evolves through interaction with other individuals in a social network. The adoption of beliefs is affected by both internal coherence and social conformity. Our framework may offer explanations for how social transitions can arise in otherwise homogeneous populations, how small numbers of zealots with highly coherent beliefs can overturn societal consensus, and how belief rigidity protects fringe groups and cults against invasion from mainstream beliefs, allowing them to persist and even thrive in larger societies. Our results suggest that strong consensus may be insufficient to guarantee social stability, that the cognitive coherence of belief-systems is vital in determining their ability to spread, and that coherent belief-systems may pose a serious problem for resolving social polarization, due to their ability to prevent consensus even under high levels of social exposure. We argue that the inclusion of cognitive factors into a social model could provide a more complete picture of collective human dynamics.
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Affiliation(s)
- Nathaniel Rodriguez
- The Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, Bloomington, Indiana, United States of America
| | - Johan Bollen
- The Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, Bloomington, Indiana, United States of America
| | - Yong-Yeol Ahn
- The Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, Bloomington, Indiana, United States of America
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13
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Verma G, Chan K, Swami A. Zealotry promotes coexistence in the rock-paper-scissors model of cyclic dominance. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:052807. [PMID: 26651744 DOI: 10.1103/physreve.92.052807] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2015] [Indexed: 06/05/2023]
Abstract
Cyclic dominance models, such as the classic rock-paper-scissors (RPS) game, have found real-world applications in biology, ecology, and sociology. A key quantity of interest in such models is the coexistence time, i.e., the time until at least one population type goes extinct. Much recent research has considered conditions that lengthen coexistence times in an RPS model. A general finding is that coexistence is promoted by localized spatial interactions (low mobility), while extinction is fostered by global interactions (high mobility). That is, there exists a mobility threshold which separates a regime of long coexistence from a regime of rapid collapse of coexistence. The key finding of our paper is that if zealots (i.e., nodes able to defeat others while themselves being immune to defeat) of even a single type exist, then system coexistence time can be significantly prolonged, even in the presence of global interactions. This work thus highlights a crucial determinant of system survival time in cyclic dominance models.
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Affiliation(s)
- Gunjan Verma
- Computational and Information Sciences Directorate, Army Research Laboratory, Adelphi, Maryland 20783, USA
| | - Kevin Chan
- Computational and Information Sciences Directorate, Army Research Laboratory, Adelphi, Maryland 20783, USA
| | - Ananthram Swami
- Computational and Information Sciences Directorate, Army Research Laboratory, Adelphi, Maryland 20783, USA
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14
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Mistry D, Zhang Q, Perra N, Baronchelli A. Committed activists and the reshaping of status-quo social consensus. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:042805. [PMID: 26565287 DOI: 10.1103/physreve.92.042805] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2015] [Indexed: 05/23/2023]
Abstract
The role of committed minorities in shaping public opinion has been recently addressed with the help of multiagent models. However, previous studies focused on homogeneous populations where zealots stand out only for their stubbornness. Here we consider the more general case in which individuals are characterized by different propensities to communicate. In particular, we correlate commitment with a higher tendency to push an opinion, acknowledging the fact that individuals with unwavering dedication to a cause are also more active in their attempts to promote their message. We show that these activists are not only more efficient in spreading their message but that their efforts require an order of magnitude fewer individuals than a randomly selected committed minority to bring the population over to a new consensus. Finally, we address the role of communities, showing that partisan divisions in the society can make it harder for committed individuals to flip the status-quo social consensus.
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Affiliation(s)
- Dina Mistry
- Laboratory for the Modeling of Biological and Socio-technical Systems, Northeastern University, Boston, Massachusetts 02115, USA
| | - Qian Zhang
- Laboratory for the Modeling of Biological and Socio-technical Systems, Northeastern University, Boston, Massachusetts 02115, USA
| | - Nicola Perra
- Laboratory for the Modeling of Biological and Socio-technical Systems, Northeastern University, Boston, Massachusetts 02115, USA
- Centre for Business Network Analysis, University of Greenwich, Park Row, London SE10 9LS, United Kingdom
| | - Andrea Baronchelli
- Department of Mathematics, City University London, London EC1V 0HB, United Kingdom
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15
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Mobilia M. Nonlinear q-voter model with inflexible zealots. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012803. [PMID: 26274221 DOI: 10.1103/physreve.92.012803] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2015] [Indexed: 06/04/2023]
Abstract
We study the dynamics of the nonlinear q-voter model with inflexible zealots in a finite well-mixed population. In this system, each individual supports one of two parties and is either a susceptible voter or an inflexible zealot. At each time step, a susceptible adopts the opinion of a neighbor if this belongs to a group of q≥2 neighbors all in the same state, whereas inflexible zealots never change their opinion. In the presence of zealots of both parties, the model is characterized by a fluctuating stationary state and, below a zealotry density threshold, the distribution of opinions is bimodal. After a characteristic time, most susceptibles become supporters of the party having more zealots and the opinion distribution is asymmetric. When the number of zealots of both parties is the same, the opinion distribution is symmetric and, in the long run, susceptibles endlessly swing from the state where they all support one party to the opposite state. Above the zealotry density threshold, when there is an unequal number of zealots of each type, the probability distribution is single-peaked and non-Gaussian. These properties are investigated analytically and with stochastic simulations. We also study the mean time to reach a consensus when zealots support only one party.
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Affiliation(s)
- Mauro Mobilia
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
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16
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Waagen A, Verma G, Chan K, Swami A, D'Souza R. Effect of zealotry in high-dimensional opinion dynamics models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022811. [PMID: 25768556 DOI: 10.1103/physreve.91.022811] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2014] [Indexed: 06/04/2023]
Abstract
Most of the work on opinion dynamics models focuses on the case of two or three opinion types. We consider the case of an arbitrary number of opinions in the mean field case of the naming game model in which it is assumed the population is infinite and all individuals are neighbors. A particular challenge of the naming game model is that the number of variables, which corresponds to the number of possible sets of opinions, grows exponentially with the number of possible opinions. We present a method for generating mean field dynamical equations for the general case of k opinions. We calculate the steady states in two important special cases in arbitrarily high dimension: the case in which there exist zealots of only one type, and the case in which there are an equal number of zealots for each opinion. We show that in these special cases a phase transition occurs at critical values p(c) of the parameter p describing the fraction of zealots. In the former case, the critical value determines the threshold value beyond which it is not possible for the opinion with no zealots to be held by more nodes than the opinion with zealots, and this point remains fixed regardless of dimension. In the latter case, the critical point p(c) is the threshold value beyond which a stalemate between all k opinions is guaranteed, and we show that it decays precisely as a lognormal curve in k.
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Affiliation(s)
- Alex Waagen
- University of California, Davis, California 95616, USA
| | - Gunjan Verma
- U. S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA
| | - Kevin Chan
- U. S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA
| | - Ananthram Swami
- U. S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA
| | - Raissa D'Souza
- University of California, Davis, California 95616, USA
- The Santa Fe Institute, Santa Fe, New Mexico 87501, USA
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17
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Boccaletti S, Bianconi G, Criado R, del Genio C, Gómez-Gardeñes J, Romance M, Sendiña-Nadal I, Wang Z, Zanin M. The structure and dynamics of multilayer networks. PHYSICS REPORTS 2014; 544:1-122. [PMID: 32834429 PMCID: PMC7332224 DOI: 10.1016/j.physrep.2014.07.001] [Citation(s) in RCA: 874] [Impact Index Per Article: 87.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 07/03/2014] [Indexed: 05/05/2023]
Abstract
In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.
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Affiliation(s)
- S. Boccaletti
- CNR - Institute of Complex Systems, Via Madonna del Piano, 10, 50019 Sesto Fiorentino, Florence, Italy
- The Italian Embassy in Israel, 25 Hamered st., 68125 Tel Aviv, Israel
| | - G. Bianconi
- School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
| | - R. Criado
- Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - C.I. del Genio
- Warwick Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
- Centre for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
- Warwick Infectious Disease Epidemiology Research (WIDER) Centre, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - J. Gómez-Gardeñes
- Institute for Biocomputation and Physics of Complex Systems, University of Zaragoza, Zaragoza, Spain
| | - M. Romance
- Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - I. Sendiña-Nadal
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
- Complex Systems Group, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
| | - Z. Wang
- Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong Special Administrative Region
- Center for Nonlinear Studies, Beijing–Hong Kong–Singapore Joint Center for Nonlinear and Complex Systems (Hong Kong) and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong Special Administrative Region
| | - M. Zanin
- Innaxis Foundation & Research Institute, José Ortega y Gasset 20, 28006 Madrid, Spain
- Faculdade de Ciências e Tecnologia, Departamento de Engenharia Electrotécnica, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
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18
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Zhao J, Liu Q, Wang X. Competitive dynamics on complex networks. Sci Rep 2014; 4:5858. [PMID: 25068622 PMCID: PMC5376163 DOI: 10.1038/srep05858] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/02/2014] [Accepted: 07/11/2014] [Indexed: 11/10/2022] Open
Abstract
We consider a dynamical network model in which two competitors have fixed and different states, and each normal agent adjusts its state according to a distributed consensus protocol. The state of each normal agent converges to a steady value which is a convex combination of the competitors' states, and is independent of the initial states of agents. This implies that the competition result is fully determined by the network structure and positions of competitors in the network. We compute an Influence Matrix (IM) in which each element characterizing the influence of an agent on another agent in the network. We use the IM to predict the bias of each normal agent and thus predict which competitor will win. Furthermore, we compare the IM criterion with seven node centrality measures to predict the winner. We find that the competitor with higher Katz Centrality in an undirected network or higher PageRank in a directed network is most likely to be the winner. These findings may shed new light on the role of network structure in competition and to what extent could competitors adjust network structure so as to win the competition.
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Affiliation(s)
- Jiuhua Zhao
- Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China
| | - Qipeng Liu
- Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China
| | - Xiaofan Wang
- Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China
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19
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Wang Z, Liu Y, Wang L, Zhang Y, Wang Z. Freezing period strongly impacts the emergence of a global consensus in the voter model. Sci Rep 2014; 4:3597. [PMID: 24398458 PMCID: PMC3884229 DOI: 10.1038/srep03597] [Citation(s) in RCA: 37] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2013] [Accepted: 12/06/2013] [Indexed: 11/24/2022] Open
Abstract
It is well known that human beings do not always change opinions or attitudes, since the duration of interaction with others usually has a significant impact on one's decision-making. Based on this observation, we introduce a freezing period into the voter model, in which the frozen individuals have a weakened opinion switching ability. We unfold the presence of an optimal freezing period, which leads to the fastest consensus, using computation simulations as well as theoretical analysis. We demonstrate that the essence of an accelerated consensus is attributed to the biased random walk of the interface between adjacent opinion clusters. The emergence of an optimal freezing period is robust against the size of the system and the number of distinct opinions. This study is instructive for understanding human collective behavior in other relevant fields.
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Affiliation(s)
- Zhen Wang
- School of Software, and Computational Social Science Laboratory, School of Innovation Experiment, Dalian University of Technology, Dalian 116621, China
| | - Yi Liu
- Department of Public Management, School of Public Administration and Law, Dalian University of Technology, Dalian 116024, China
| | - Lin Wang
- 1] Adaptive Networks and Control Laboratory, Department of Electronic Engineering, Fudan University, Shanghai 200433, China [2] Centre for Chaos and Complex Networks, Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China
| | - Yan Zhang
- Adaptive Networks and Control Laboratory, Department of Electronic Engineering, Fudan University, Shanghai 200433, China
| | - Zhen Wang
- 1] Department of Physics, Hong Kong Baptist University, Hong Kong SAR, China [2] Center for Nonlinear Studies, and the Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Hong Kong), Hong Kong Baptist University, Hong Kong SAR, China [3] Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China
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20
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On the effect of heterogeneity in stochastic interacting-particle systems. Sci Rep 2013; 3:1189. [PMID: 23378920 PMCID: PMC3561626 DOI: 10.1038/srep01189] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2012] [Accepted: 01/09/2013] [Indexed: 12/03/2022] Open
Abstract
We study stochastic particle systems made up of heterogeneous units. We introduce a general framework suitable to analytically study this kind of systems and apply it to two particular models of interest in economy and epidemiology. We show that particle heterogeneity can enhance or decrease the size of the collective fluctuations depending on the system, and that it is possible to infer the degree and the form of the heterogeneity distribution in the system by measuring only global variables and their fluctuations. Our work shows that, in some cases, heterogeneity among the units composing a system can be fully taken into account without losing analytical tractability.
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