×

Dynamics of opinions with social biases. (English) Zbl 1429.91258

Summary: This paper aims to provide a systemic analysis to social opinion dynamics subject to individual biases. As a generalization of the classical DeGroot social interactions, defined by linearly coupled dynamics of peer opinions that evolve over time, biases add to state-dependent edge weights and therefore lead to highly nonlinear network dynamics. Previous studies have dealt with convergence and stability analysis of such systems for a few specific initial node opinions and network structures, and here we focus on how individual biases affect social equilibria and their stabilities. Two categories of equilibria, namely the boundary and interior equilibria, are defined. For a few fundamental network structures, some important interior network equilibria are presented explicitly for a wide range of system parameters, which are shown to be locally unstable in general. Particularly, the interval centroid is proven to be unstable regardless of the bias level and the network topologies. Next, we prove that when the initial network opinions are polarized towards one side of the state space, node biases will drive the opinion evolution to the corresponding interval boundaries. Such polarization attraction effect continues to hold under even directed and switching network structures. Finally, a number of numerical examples are provided to validate our study and advance the understanding of the nonlinearity inherited within the biased opinion evolution.

MSC:

91D30 Social networks; opinion dynamics
93C10 Nonlinear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acemoglu, D.; Como, G.; Fagnani, F.; Ozdaglar, A., Opinion fluctuations and disagreement in social networks, Mathematics of Operations Research, 38, 1, 1-27 (2013) · Zbl 1297.91130
[2] Altafini, C., Dynamics of opinion forming in structurally balanced social networks, PLoS One, 7, 6, Article e38135 pp. (2012)
[3] Altafini, C., Consensus problems on networks with antagonistic interactions, IEEE Transactions on Automatic Control, 58, 4, 935-946 (2013) · Zbl 1369.93433
[4] Bauso, D.; Giarre, L.; Pesenti, R., Non-linear protocols for optimal distributed consensus in networks of dynamic agents, Systems & Control Letters, 55, 11, 918-928 (2006) · Zbl 1111.68009
[5] Bhawalkar, K., Gollapudi, S., & Munagala, K. (2013). Coevolutionary opinion formation games. In Proceedings of the 45th annual ACM symposium on theory of computing; Bhawalkar, K., Gollapudi, S., & Munagala, K. (2013). Coevolutionary opinion formation games. In Proceedings of the 45th annual ACM symposium on theory of computing
[6] Bindel, D.; Kleinberg, J.; Oren, S., How bad is forming your own opinion, Games and Economic Behavior, 24, 8-265 (2015) · Zbl 1318.91156
[7] Blondel, V. D., Hendrickx, J. M., Olshevsky, A., & Tsitsiklis, J. N. (2005). Convergence in multiagent coordination, consensus, and flocking. In 44th IEEE conference on decision and control; Blondel, V. D., Hendrickx, J. M., Olshevsky, A., & Tsitsiklis, J. N. (2005). Convergence in multiagent coordination, consensus, and flocking. In 44th IEEE conference on decision and control
[8] Blondel, V. D.; Hendrickx, J. M.; Tsitsiklis, J. N., On Krause’s multi-agent consensus model with state-dependent connectivity, IEEE Transactions on Automatic Control, 54, 11, 2586-2597 (2009) · Zbl 1367.93426
[9] Cao, M.; Morse, A. S.; Anderson, B. D.O., Reaching a consensus in a dynamically changing environment: A graphical approach, SIAM Journal on Control and Optimization, 47, 2, 575-600 (2008) · Zbl 1157.93514
[10] Centola, D., An experimental study of homophily in the adoption of health behavior, Science, 334, 6060, 1269-1272 (2011)
[11] Cox, D.; Little, J.; O’shea, D., Ideals, varieties, and algorithms (Vol. 3) (2007), Springer: Springer New York
[12] Dandekar, P.; Goel, A.; Lee, D. T., Biased assimilation, homophily, and the dynamics of polarization, Proceedings of the National Academy of Sciences, 110, 15, 5791-5796 (2013) · Zbl 1292.91147
[13] DeGroot, M. H., Reaching a consensus, Journal of the American Statistical Association, 69, 345, 118-121 (1974) · Zbl 0282.92011
[14] DeMarzo, P. M.; Vayanos, D.; Zwiebel, J., Persuasion bias, social influence, and unidimensional opinions, Quarterly Journal of Economics, 118, 3, 909-968 (2003) · Zbl 1069.91093
[15] Easley, D.; Kleinberg, J., Networks, crowds, and markets: Reasoning about a highly connected world (2010), Cambridge University Press · Zbl 1205.91007
[16] Friedkin, N. E.; Johnsen, E. C., Social influence and opinions, Journal of Mathematical Sociology, 15, 3-4, 193-206 (1990) · Zbl 0712.92025
[17] Friedkin, N. E.; Proskurnikov, A. V.; Tempo, R.; Parsegov, S. E., Network science on belief system dynamics under logic constraints, Science, 354, 6310, 321-326 (2016) · Zbl 1355.91060
[18] Golub, B.; Jackson, M. O., Naive learning in social networks and the wisdom of crowds, American Economic Journal: Microeconomics, 2, 1, 112-149 (2010)
[19] Hegselmann, R.; Krause, U., Opinion dynamics and bounded confidence models, analysis, and simulation, Journal of Artificial Societies and Social Simulation, 5, 3, 1-33 (2002)
[20] Jackson, M. O., Social and economic networks (2010), Princeton University Press · Zbl 1203.91001
[21] Jadbabaie, A.; Lin, J.; Morse, A. S., Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, 48, 6, 988-1001 (2003) · Zbl 1364.93514
[22] Khalil, H. K., Nonlinear Systems (2002), Prentice Hall: Prentice Hall New Jewsey
[23] Lawrence, E.; Sides, J.; Farrell, H., Self-segregation or deliberation? Blog readership, participation, and polarization in American politics, Perspectives on Politics, 8, 1, 141-157 (2010)
[24] Leonard, N. E., Multi-agent system dynamics: Bifurcation and behavior of animal groups, Annual Reviews in Control, 38, 2, 171-183 (2014)
[25] Li, L.; Scaglione, A.; Swami, A.; Zhao, Q., Consensus, polarization and clustering of opinions in social networks, IEEE Journal on Selected Areas in Communications, 31, 6, 1072-1083 (2013)
[26] Lin, Z.; Francis, B.; Maggiore, M., State agreement for continuous-time coupled nonlinear systems, SIAM Journal on Control and Optimization, 46, 1, 288-307 (2007) · Zbl 1141.93032
[27] Lord, C. G.; Ross, L.; Lepper, M. R., Biased assimilation and attitude polarization: The effects of prior theories on subsequently considered evidence, Journal of Personality and Social Psychology, 37, 11, 2098-2109 (1979)
[28] Mas, M.; Flache, A.; Helbing, D., Individualization as driving force of clustering phenomena in humans, PLoS Computational Biology, 6, 10, e1000959 (2010)
[29] McCarty, N.; Poole, K. T.; Rosenthal, H., Polarized America: The dance of ideology and unequal riches (2016), MIT Press
[30] Miller, A. G.; Mchoskey, J. W.; Bane, C. M.; Dowd, T. G., The attitude polarization phenomenon: Role of response measure, attitude extremity, and behavioral consequences of reported attitude change, Journal of Personality and Social Psychology, 64, 4, 561-574 (1993)
[31] Moreau, L., Stability of multiagent systems with time-dependent communication links, IEEE Transactions on Automatic Control, 50, 2, 169-182 (2005) · Zbl 1365.93268
[32] Scott, J., Social network analysis (2017), Sage
[33] Semonsen, J.; Griffin, C.; Squicciarini, A.; Rajtmajer, S., Opinion dynamics in the presence of increasing agreement pressure, IEEE Transactions on Cybernetics, 1-9 (2018)
[34] Shi, G.; Anderson, B. D.O.; Johansson, K. H., Consensus over random graph processes: Network Borel-Cantelli lemmas for almost sure convergence, IEEE Transactions on Information Theory, 61, 10, 5690-5707 (2015) · Zbl 1359.94283
[35] Shi, G.; Johansson, M.; Johansson, K. H., How agreement and disagreement evolve over random dynamic networks, IEEE Journal on Selected Areas in Communications, 31, 6, 1061-1071 (2013)
[36] Shi, G.; Proutiere, A.; Johansson, M.; Baras, J. S.; Johansson, K. H., The evolution of beliefs over signed social networks, Operations Research, 64, 3, 585-604 (2016) · Zbl 1348.91239
[37] Taber, C. S.; Lodge, M., Motivated skepticism in the evaluation of political beliefs, American Journal of Political Science, 50, 3, 755-769 (2006)
[38] Tsitsiklis, J. N., Problems in decentralized decision making and computation (1984), Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology: Dept. of Electrical Engineering and Computer Science, Massachusetts Institute of Technology Boston, MA, (Ph.D. thesis)
[39] Tsitsiklis, J. N.; Bertsekas, D.; Athans, M., Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Transactions on Automatic Control, 31, 9, 803-812 (1986) · Zbl 0602.90120
[40] Xiao, L.; Boyd, S., Fast linear iterations for distributed averaging, Systems & Control Letters, 53, 1, 65-78 (2004) · Zbl 1157.90347
[41] Yildiz, E.; Ozdaglar, A.; Acemoglu, D.; Saberi, A.; Scaglione, A., Binary opinion dynamics with stubborn agents, ACM Transactions on Economics and Computation, 1, 4 (2013), Art. 19
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.