Broken detailed balance and non-equilibrium dynamics in noisy social learning models. (English) Zbl 1478.82005
Summary: We propose new Degroot-type social learning models with noisy feedback in continuous time. Unlike the standard Degroot framework, noisy information frameworks destroy consensus formation. On the other hand, noisy opinion dynamics converge to the equilibrium distribution that encapsulates correlations among agents’ opinions. Interestingly, such an equilibrium distribution is also a non-equilibrium steady state (NESS) with a non-zero probabilistic current loop. Thus, noisy information source leads to a NESS at long times that encodes persistent correlated opinion dynamics of learning agents. Our model provides a simple realization of NESS in the context of social learning. Other phenomena such as synchronization of opinions when agents are subject to a common noise are also studied.
MSC:
| 82B31 | Stochastic methods applied to problems in equilibrium statistical mechanics |
| 92D99 | Genetics and population dynamics |
| 91A99 | Game theory |
| 82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |
| 91B80 | Applications of statistical and quantum mechanics to economics (econophysics) |
Keywords:
DeGroot learning; social learning; non-equilibrium steady state; broken detailed balance; synchronization of opinionsReferences:
| [1] | Onnela, J.-P.; Reed-Tsochas, F., Spontaneous emergence of social influence in online systems, Proc. Natl. Acad. Sci., 107, 43, 18375-18380 (2010) |
| [2] | Schinckus, C., Ising model, econophysics and analogies, Physica A, 508, 95-103 (2018) |
| [3] | Leskovec, J.; McGlohon, M.; Faloutsos, C.; Glance, N.; Hurst, M., Patterns of cascading behavior in large blog graphs, (Proceedings of the 2007 SIAM International Conference on Data Mining (2007), SIAM), 551-556 |
| [4] | M.P. Simmons, L.A. Adamic, E. Adar, Memes online: Extracted, subtracted, injected, and recollected, in: Fifth International AAAI Conference on Weblogs and Social Media, 2011, pp. 353-360. |
| [5] | Fernández-Gracia, J.; Suchecki, K.; Ramasco, J. J.; San Miguel, M.; Eguíluz, V. M., Is the voter model a model for voters?, Phys. Rev. Lett., 112, 15, Article 158701 pp. (2014) |
| [6] | Das, A.; Gollapudi, S.; Munagala, K., Modeling opinion dynamics in social networks, (Proceedings of the 7th ACM International Conference on Web Search and Data Mining (2014), ACM), 403-412 |
| [7] | Masuda, N.; Kawamura, Y.; Kori, H., Collective fluctuations in networks of noisy components, New J. Phys., 12, 9, Article 093007 pp. (2010) |
| [8] | Clifford, P.; Sudbury, A., A model for spatial conflict, Biometrika, 60, 3, 581-588 (1973) · Zbl 0272.60072 |
| [9] | Holley, R. A.; Liggett, T. M., Ergodic theorems for weakly interacting infinite systems and the voter model, Ann. Probab., 3, 4, 643-663 (1975) · Zbl 0367.60115 |
| [10] | Castellano, C.; Fortunato, S.; Loreto, V., Statistical physics of social dynamics, Rev. Modern Phys., 81, 2, 591-646 (2009) |
| [11] | Sood, V.; Antal, T.; Redner, S., Voter models on heterogeneous networks, Phys. Rev. E, 77, 4, Article 041121 pp. (2008) |
| [12] | Liggett, T., Interacting Particle Systems (1985), Springer-Verlag · Zbl 0559.60078 |
| [13] | Krapivsky, P.; Redner, S.; Ben-Naim, E., A Kinetic View of Statistical Physics (2010), Cambridge University Press · Zbl 1235.82040 |
| [14] | Chakraborti, A.; Toke, I. M.; Patriarca, M.; Abergel, F., Econophysics review: I. Empirical facts, Quant. Finance, 11, 7, 991-1012 (2011) |
| [15] | Chakraborti, A.; Toke, I. M.; Patriarca, M.; Abergel, F., Econophysics review: II. Agent-based models, Quant. Finance, 11, 7, 1013-1041 (2011) |
| [16] | Abergel, F.; Chakrabarti, B. K.; Chakraborti, A.; Mitra, M., Econophysics of Order-Driven Markets (2011), Springer Science & Business Media · Zbl 1303.91007 |
| [17] | Stanley, H. E.; Afanasyev, V.; Amaral, L. A.N.; Buldyrev, S.; Goldberger, A.; Havlin, S.; Leschhorn, H.; Maass, P.; Mantegna, R. N.; Peng, C. K., Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics, Phys.-Section A, 224, 1, 302-321 (1996) |
| [18] | Degroot, M. H., Reaching a consensus, J. Amer. Statist. Assoc., 69, 345, 118-121 (1974) · Zbl 0282.92011 |
| [19] | Masuda, N.; Porter, M. A.; Lambiotte, R., Random walks and diffusion on networks, Phys. Rep., 716, 1-58 (2017) · Zbl 1377.05180 |
| [20] | Noorazar, H.; Vixie, K. R.; Telebanpour, A.; Hu, Y., From classical to modern opinion dynamics, Internat. J. Modern Phys. C, 31, 07, Article 2050101 pp. (2020) |
| [21] | Noorazar, H., Recent advances in opinion propagation dynamics: a 2020 survey, Eur. Phys. J. Plus, 135, 521 (2020) |
| [22] | Golub, B.; Sadler, E., (Learning in Social Networks. Learning in Social Networks, The Oxford Handbook of the Economics of Networks (2016), Oxford University Press) |
| [23] | Golub, B.; Jackson, M. O., Naive learning in social networks and the wisdom of crowds, Am. Econ. J. Microecon., 2, 1, 112-149 (2010) |
| [24] | Mossel, E.; Tamuz, O., Opinion exchange dynamics, Probab. Surv., 14, 155-204 (2017) · Zbl 1376.91131 |
| [25] | Liu, Z.; Ma, J.; Zeng, Y.; Yang, L.; Huang, Q.; Wu, H., On the control of opinion dynamics in social networks, Physica A, 409, 183-198 (2014) · Zbl 1402.91635 |
| [26] | Deffuant, G.; Neau, D.; Amblard, F.; Weisbuch, G., Mixing beliefs among interacting agents, Adv. Complex Syst., 03, 01n04, 87-98 (2000) |
| [27] | Hegselmann, R.; social, U. K., Opinion dynamics and bounded confidence models, analysis, and simulation, J. Artif. Soc. Soc. Simul. (2002) |
| [28] | Lorenz, J., Continuous opinion dynamics under bounded confidence: A survey, Internat. J. Modern Phys. C, 18, 12, 1819-1838 (2007) · Zbl 1151.91076 |
| [29] | Becker, J.; Brackbill, D.; Centola, D., Network dynamics of social influence in the wisdom of crowds, Proc. Natl. Acad. Sci., 114, 26, E5070-E5076 (2017) |
| [30] | Chandrasekhar, A. G.; Larreguy, H.; Xandri, J. P., Testing Models of Social Learning on Networks: Evidence from a Lab Experiment in the FieldTech. rep. (2015), National Bureau of Economic Research |
| [31] | Askarzadeh, Z.; Fu, R.; Halder, A.; Chen, Y.; Georgiou, T. T., Stability theory of stochastic models in opinion dynamics, IEEE Trans. Automat. Control, 65, 2, 522-533 (2019) · Zbl 1533.91390 |
| [32] | Bolley, F.; Gentil, I.; Guillin, A., Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263, 8, 2430-2457 (2012) · Zbl 1253.35183 |
| [33] | Vaidya, T.; Murguia, C.; Piliouras, G., Learning agents in financial markets: Consensus dynamics on volatility, (Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems (2018), International Foundation for Autonomous Agents and Multiagent Systems), 2106-2108 |
| [34] | Foucault, T.; Pagano, M.; Roell, A.; Röell, A., Market Liquidity: Theory, Evidence, and Policy (2013), Oxford University Press |
| [35] | Suhov, Y.; Kelbert, M., Probability and Statistics by Example: Volume 2, Markov Chains: A Primer in Random Processes and Their Applications, Vol. 2 (2008), Cambridge University Press · Zbl 1184.62002 |
| [36] | Evans, L. C., An Introduction to Stochastic Differential Equations, Vol. 82 (2012), American Mathematical Soc. |
| [37] | Yuan, R.; Ao, P., Beyond itô versus stratonovich, J. Stat. Mech. Theory Exp., 2012, 07, P07010 (2012) · Zbl 1456.60168 |
| [38] | Mannella, R.; McClintock, P. V., Itô versus Stratonovich: 30 years later, Fluct. Noise Lett., 11, 01, Article 1240010 pp. (2012) |
| [39] | Galayda, S.; Barany, E., Stochastic differential equation derivation: Comparison of the Markov method versus the additive method, Physica A, 391, 20, 4564-4574 (2012) |
| [40] | Föllmer, H.; Schweizer, M., A microeconomic approach to diffusion models for stock prices, Math. Finance, 3, 1, 1-23 (1993) · Zbl 0884.90027 |
| [41] | Horst, U., Financial price fluctuations in a stock market model with many interacting agents, Econom. Theory, 25, 4, 917-932 (2005) · Zbl 1134.91426 |
| [42] | Henkel, C., An Agent Behavior Based Model for Diffusion Price Processes (2017), lmu, (Ph.D. thesis) |
| [43] | Henkel, C., From quantum mechanics to finance: Microfoundations for jumps, spikes and high volatility phases in diffusion price processes, Physica A, 469, 447-458 (2017) · Zbl 1400.91681 |
| [44] | Pakkanen, M. S., Microfoundations for diffusion price processes, Math. Financ. Econ., 3, 2, 89-114 (2010) · Zbl 1255.91441 |
| [45] | Huang, G.; Jansen, H. M.; Mandjes, M.; Spreij, P.; De Turck, K., Markov-modulated ornstein-uhlenbeck processes, Adv. Appl. Probab., 48, 1, 235-254 (2016) · Zbl 1337.60191 |
| [46] | da Fonseca, R. C.; Figueiredo, A.; de Castro, M. T.; Mendes, F. M., Generalized Ornstein-Uhlenbeck process by Doob’s theorem and the time evolution of financial prices, Physica A, 392, 7, 1671-1680 (2013) |
| [47] | Ramsza, M.; Seymour, R. M., Fictitious play in an evolutionary environment, Games Econom. Behav., 68, 1, 303-324 (2010) · Zbl 1197.91048 |
| [48] | Lima, L. S.; Oliveira, S.; Abeilice, A.; Melgaço, J., Breaks down of the modeling of the financial market with addition of non-linear terms in the Itô stochastic process, Physica A, 526, Article 120932 pp. (2019) · Zbl 07566419 |
| [49] | Kessler, D. A.; Burov, S., Stochastic maps, continuous approximation, and stable distribution, Phys. Rev. E, 96, 4, Article 042139 pp. (2017) |
| [50] | Pavliotis, G. A., Stochastic Processes and Applications (2011), Springer |
| [51] | Gardiner, C., Handbook of Stochastic Processes. Handbook of Stochastic Processes, Springer-Verlag, Berlin (1985) |
| [52] | Lasota, A.; Mackey, M. C., Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Vol. 97 (2013), Springer Science & Business Media |
| [53] | Khasminskii, R., Stochastic Stability of Differential Equations, Vol. 66 (2011), Springer Science & Business Media |
| [54] | Liu, Y.-Y.; Slotine, J.-J.; Barabási, A.-L., Controllability of complex networks, Nature, 473, 7346, 167-173 (2011) |
| [55] | Karatzas, I.; Shreve, S. E., Brownian motion, (Brownian Motion and Stochastic Calculus (1998), Springer), 47-127 |
| [56] | Priola, E.; Zabczyk, J., Densities for Ornstein-Uhlenbeck processes with jumps, Bull. Lond. Math. Soc., 41, 1, 41-50 (2008) · Zbl 1166.60034 |
| [57] | Elliott, D. L., A consequence of controllability, J. Differential Equations, 10, 2, 364-370 (1971) · Zbl 0216.27502 |
| [58] | Kashima, K., Noise response data reveal novel controllability Gramian for nonlinear network dynamics, Sci. Rep., 6, 27300 (2016) |
| [59] | Brockett, R., The early days of geometric nonlinear control, Automatica, 50, 9, 2203-2224 (2014) · Zbl 1297.93052 |
| [60] | Whalen, A. J.; Brennan, S. N.; Sauer, T. D.; Schiff, S. J., Observability and controllability of nonlinear networks: The role of symmetry, Phys. Rev. X, 5, 1, Article 011005 pp. (2015) |
| [61] | Mellor, A.; Mobilia, M.; Zia, R. K.P., Characterization of the nonequilibrium steady state of a heterogeneous nonlinear q-voter model with zealotry, EPL (Europhysics Letters), 113, 4, 48001 (2016) |
| [62] | Mellor, A.; Mobilia, M.; Zia, R., Heterogeneous out-of-equilibrium nonlinear q-voter model with zealotry, Phys. Rev. E, 95, 1, Article 012104 pp. (2017) |
| [63] | Zia, R. K.P.; Schmittmann, B., Probability currents as principal characteristics in the statistical mechanics of non-equilibrium steady states, J. Stat. Mech. Theory Exp., 2007, 07, P07012 (2007) |
| [64] | Risken, H., Fokker-Planck Equation (1984), Springer · Zbl 0546.60084 |
| [65] | Van Kampen, N. G., Stochastic Processes in Physics and Chemistry, Vol. 1 (1992), Elsevier |
| [66] | Gnesotto, F. S.; Mura, F.; Gladrow, J.; Broedersz, C. P., Broken detailed balance and non-equilibrium dynamics in living systems: a review, Rep. Progr. Phys., 81, 6, Article 066601 pp. (2018) |
| [67] | Mancois, V.; Marcos, B.; Viot, P.; Wilkowski, D., Two-temperature Brownian dynamics of a particle in a confining potential, Phys. Rev. E, 97, Article 052121 pp. (2018), https://link.aps.org/doi/10.1103/PhysRevE.97.052121 |
| [68] | Zhang, K.; Wang, J., Landscape and flux theory of non-equilibrium open economy, Physica A, 482, 189-208 (2017) · Zbl 1495.91072 |
| [69] | Liverpool, T. B., Steady-state distributions and nonsteady dynamics in nonequilibrium systems, Phys. Rev. E, 101, 4, Article 042107 pp. (2020) |
| [70] | McKane, A. J.; Newman, T. J., Predator-prey cycles from resonant amplification of demographic stochasticity, Phys. Rev. Lett., 94, 21, Article 218102 pp. (2005) |
| [71] | Eab, C. H.; Lim, S., Ornstein-Uhlenbeck process with fluctuating damping, Physica A, 492, 790-803 (2018) · Zbl 1514.82156 |
| [72] | Zhang, Y.; Chen, F., Stochastic stability of fractional Fokker-Planck equation, Physica A, 410, 35-42 (2014) · Zbl 1395.60067 |
| [73] | Ren, F.-Y.; Liang, J.-R.; Qiu, W.-Y.; Xiao, J.-B., Asymptotic behavior of a fractional Fokker-Planck-type equation, Physica A, 373, 165-173 (2007) |
| [74] | Li, Z.; Zhan, W.; Xu, L., Stochastic differential equations with time-dependent coefficients driven by fractional Brownian motion, Physica A, 530, Article 121565 pp. (2019) · Zbl 07568827 |
| [75] | Gajda, J.; Wyłomańska, A., Fokker-Planck type equations associated with fractional Brownian motion controlled by infinitely divisible processes, Physica A, 405, 104-113 (2014) · Zbl 1395.60045 |
| [76] | Clauset, A.; Newman, M. E.J.; Moore, C., Finding community structure in very large networks, Phys. Rev. E, 70, Article 066111 pp. (2004) |
| [77] | Zhang, X.; Moore, C.; Newman, M. E., Random graph models for dynamic networks, Eur. Phys. J. B, 90, 10, 200 (2017) |
| [78] | Weber, D.; Theisen, R.; Motsch, S., Deterministic versus stochastic consensus dynamics on graphs, J. Stat. Phys., 176, 1, 40-68 (2019) · Zbl 1422.91622 |
| [79] | Ieda, M.; Shiino, M., Modeling asset price processes based on mean-field framework, Phys. Rev. E, 84, 6, Article 066105 pp. (2011) |
| [80] | Jabin, P.-E.; Wang, Z., Mean field limit for stochastic particle systems, (Active Particles, Vol. 1 (2017), Springer), 379-402 |
| [81] | Bindel, D.; Kleinberg, J.; Oren, S., How bad is forming your own opinion?, Games Econom. Behav., 92, 248-265 (2015), http://www.sciencedirect.com/science/article/pii/S0899825614001122 · Zbl 1318.91156 |
| [82] | Ghaderi, J.; Srikant, R., Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate, Automatica, 50, 12, 3209-3215 (2014) · Zbl 1309.93010 |
| [83] | Thomas, P. J.; Lindner, B., Phase descriptions of a multidimensional Ornstein-Uhlenbeck process, Phys. Rev. E, 99, 6, Article 062221 pp. (2019) |
| [84] | Villani, C., Optimal Transport. Old and New (2009), Springer-Verlag: Springer-Verlag Berlin · Zbl 1156.53003 |
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