The Annals of Applied Probability

Scaling limits for continuous opinion dynamics systems

Giacomo Como and Fabio Fagnani

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Scaling limits are analyzed for stochastic continuous opinion dynamics systems, also known as gossip models. In such models, agents update their vector-valued opinion to a convex combination (possibly agent- and opinion-dependent) of their current value and that of another observed agent. It is shown that, in the limit of large agent population size, the empirical opinion density concentrates, at an exponential probability rate, around the solution of a probability-measure-valued ordinary differential equation describing the system’s mean-field dynamics. Properties of the associated initial value problem are studied. The asymptotic behavior of the solution is analyzed for bounded-confidence opinion dynamics, and in the presence of an heterogeneous influential environment.

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Ann. Appl. Probab., Volume 21, Number 4 (2011), 1537-1567.

First available in Project Euclid: 8 August 2011

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 91D30: Social networks 93A15: Large scale systems

Multi-agent systems social networks opinion dynamics bounded confidence scaling limits probability-measure-valued ODEs


Como, Giacomo; Fagnani, Fabio. Scaling limits for continuous opinion dynamics systems. Ann. Appl. Probab. 21 (2011), no. 4, 1537--1567. doi:10.1214/10-AAP739.

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  • [1] Acemoglu, D., Como, G., Fagnani, F. and Ozdaglar, A. (2010). Opinion fluctuations and disagreement in social networks. Unpublished manuscript. Available at
  • [2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd ed. Wiley, Hoboken, NJ.
  • [3] Ambrosio, L., Gigli, N. and Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, Basel.
  • [4] Axelrod, R. (1997). The dissemination of culture. J. of Conflict Resolution 42 203–226.
  • [5] Ben-Naim, E., Krapivsky, P. L. and Redner, S. (2003). Bifurcation and patterns in compromise processes. Phys. D 183 190–204.
  • [6] Blondel, V. D., Hendrickx, J. M. and Tsitsiklis, J. N. (2009). Continuous-time average-preserving opinion dynamics with opinion-dependent communications. Unpublished manuscript.
  • [7] Blondel, V. D., Hendrickx, J. M. and Tsitsiklis, J. N. (2009). On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Trans. Automat. Control 54 2586–2597.
  • [8] Borkar, V. S. (1995). Probability Theory: An Advanced Course. Springer, New York.
  • [9] Boyd, S., Ghosh, A., Prabhakar, B. and Shah, D. (2006). Randomized gossip algorithms. IEEE Trans. Inform. Theory 52 2508–2530.
  • [10] Canuto, C., Fagnani, F. and Tilli, P. (2008). A Eulerian approach to the analysis of rendez-vous algorithms. In Proc. of 2008 IFAC Conf., Seoul, Korea, July 611 9039–9044. Available at
  • [11] Castellano, C., Fortunato, S. and Loreto, V. (2009). Statistical physics of social dynamics. Rev. Mod. Phys. 81 591–646.
  • [12] Cucker, F. and Smale, S. (2007). Emergent behavior in flocks. IEEE Trans. Automat. Control 52 852–862.
  • [13] Cucker, F. and Smale, S. (2007). On the mathematics of emergence. Jpn. J. Math. 2 197–227.
  • [14] Deffuant, G., Neau, D., Amblard, F. and Weisbuch, G. (2000). Mixing beliefs among interacting agents. Adv. Complex Syst. 3 87–98.
  • [15] Deffuant, G., Neau, D., Amblard, F. and Weisbuch, G. (2004). Modelling group opinion shift to extreme: The smooth bounded confidence model. In Proc. Europ. Social Simulation Assoc. Conf. Valladolid, Spain.
  • [16] Edelstein-Keshet, L. and Ermentrout, G. B. (1990). Models for contact-mediated pattern formation: Cells that form parallel arrays. J. Math. Biol. 29 33–58.
  • [17] Fagnani, F. and Zampieri, S. (2008). Asymmetric randomized gossip algorithms for consensus. In Proc. of 2008 IFAC Conf., Seoul, Korea, July 6-11 9052–9056. Available at World_Congress__2008/index.html.
  • [18] Fagnani, F. and Zampieri, S. (2008). Randomized consensus algorithms over large scale networks. IEEE J. Select. Aereas Commun. 26 634–649.
  • [19] Fortunato, S. (2005). On the consensus threshold for the opinion dynamics of Krause–Hegselmann. Int. J. Mod. Phys. 16 259–270.
  • [20] Fournier, N. and Méléard, S. (2004). A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann. Appl. Probab. 14 1880–1919.
  • [21] Ha, S.-Y. and Liu, J.-G. (2009). A simple proof of the Cucker–Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7 297–325.
  • [22] Krause, U. (2000). A discrete nonlinear and non-autonomous model of consensus formation. In Communications in Difference Equations (Poznan, 1998) (S. Elaydi, G. Ladas, J. Popenda and J. Rakowski, eds.) 227–236. Gordon and Breach, Amsterdam.
  • [23] Lorenz, J. (2005). A stabilization theorem for dynamics of continuous opinions. Phys. A 355 217–223.
  • [24] Lorenz, J. (2007). Continuous opinion dynamics under bounded confidence: A survey. Internat. J. Modern Phys. C 18 1819–1838.
  • [25] Lorenz, J. (2007). Repeated averaging and bounded confidence? Modeling analysis and simulation of continuous opinion dynamics. Ph.D. thesis, Univ. Bremen. Available at
  • [26] Lorenz, J. (2010). Heterogeneous bounds of confidence: Meet, discuss and find consensus! Complexity 15 43–52.
  • [27] Remenik, D. (2009). Limit theorems for individual-based models in economics and finance. Stochastic Process. Appl. 119 2401–2435.
  • [28] Shah, D. (2008). Gossip algorithms. Foundations and Trends in Networking 3 1–125.
  • [29] Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analysis: Queues, Communications, and Computing. Chapman & Hall, London.
  • [30] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [31] Weisbuch, G., Deffuant, G., Amblard, F. and Nadal, J. P. (2002). Meet, discuss, and segregate! Complexity 7 55–63.
  • [32] Wormald, N. C. (1995). Differential equations for random processes and random graphs. Ann. Appl. Probab. 5 1217–1235.