We take a logical approach to threshold models, used to study the diffusion of opinions, new technologies, infections, or behaviors in social networks. Threshold models consist of a network graph of agents connected by a social relationship and a threshold value which regulates the diffusion process. Agents adopt a new behavior/product/opinion when the proportion of their neighbors who have already adopted it meets the threshold. Under this diffusion policy, threshold models develop dynamically towards a guaranteed fixed point. We construct a minimal dynamic propositional logic to describe the threshold dynamics and show that the logic is sound and complete. We then extend this framework with an epistemic dimension and investigate how information about more distant neighbors’ behavior allows agents to anticipate changes in behavior of their closer neighbors. Overall, our logical formalism captures the interplay between the epistemic and social dimensions in social networks.
Apt, K. R., and E. Markakis, Diffusion in Social Networks with Competing Products, in G. Persiano, (ed.), SAGT 2011, LNCS 6982, Springer, 2011, pp. 212–223.
Aumann, R. J., Agreeing to disagree, Annals of Statistics 4: 1236–1239, 1976.
Baltag, A., H. van Ditmarsch, and L. Moss, Epistemic logic and information update, in P. Adriaans, and J. van Benthem, (eds.), Handbook on the Philosophy of Information, Elsevier, Amsterdam, 2008.
Baltag, A., L. Moss, and S. Solecki, The logic of public announcements, common knowledge and private suspicions, in Proceedings of TARK’98 (Seventh Conference on Theoretical Aspects of Rationality and Knowledge), Morgan Kaufmann Publishers, 1998, pp. 43–56.
Baltag, A., and S. Smets, A qualitative theory of dynamic interactive belief revision, in G. Bonanno, W. van der Hoek, and M. Wooldridge, (eds.), Logic and the Foundations of Game and Decision Theory (LOFT 7), Texts in Logic and Games, Vol. 3, Amsterdam University Press, 2008, pp. 9–58.
van Benthem, J., Logical Dynamics of Information and Interaction, Cambridge University Press, Cambridge, 2011.
van Benthem, J., Oscillations, logic, and dynamical systems, in S. Ghosh, and J. Szymanik, (eds.), The Facts Matter. Essays on Logic and Cognition in Honour of Rineke Verbrugge, College Publications, London, 2015, pp. 9–22.
Christoff, Z., Dynamic Logics of Networks: Information Flow and the Spread of Opinion, Ph.D. thesis, Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands, 2016. ILLC Dissertation Series DS-2016-02.
Christoff, Z., and D. Grossi, Stability in binary opinion diffusion, in A. Baltag, J. Seligman, and T. Yamada, (eds.), Logic, Rationality, and Interaction: 6th International Workshop, LORI 2017, Sapporo, Japan, September 11–14, 2017, Proceedings, Springer, Berlin, 2017, pp. 166–180.
Christoff, Z., and J. U. Hansen, A two-tiered formalization of social influence, in D. Grossi, O. Roy, and H. Huang, (eds.), Logic, Rationality, and Interaction, Vol. 8196 of Lecture Notes in Computer Science, Springer, Berlin, 2013, pp. 68–81.
Christoff, Z., and J. U. Hansen, A logic for diffusion in social networks, Journal of Applied Logic 13(1): 48–77, 2015.
Christoff, Z., J. U. Hansen, and C. Proietti, Reflecting on social influence in networks, Journal of Logic, Language and Information, 25(3-4): 299–333, 2016.
Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, second edn., Cambridge University Press, Cambridge, 2002.
van Ditmarsch, H., W. van der Hoek, and B. Kooi, Dynamic Epistemic Logic, Springer, New York, 2008.
Easley, D., and J. Kleinberg, Networks, Crowds, and Markets, Cambridge University Press, Cambridge, 2010.
Fitting, M., L. Thalmann, and A. Voronkov, Term-modal logics, Studia Logica 69: 133–169, 2001.
Goyal, S., Interaction structure and social change, Journal of Institutional and Theoretical Economics 152(3): 472–494, 1996.
Granovetter, M., Threshold models of collective behavior, American Journal of Sociology 83(6): 1420–1443, 1978.
Hansen, P. G., V. F. Hendricks, and R. K. Rendsvig, Infostorms, Metaphilosophy 44(3): 301–326, 2013.
Kempe, D., J. Kleinberg, and É. Tardos, Maximizing the spread of influence through a social network, in KDD 03: Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, New York, NY, 2003, pp. 137–146.
Klein, D., and R. K. Rendsvig, Convergence, continuity and recurrence in dynamic epistemic logic, in A. Baltag, and J. Seligman, (eds.), Logic, Rationality, and Interaction (LORI 2017, Sapporo), Vol. 10455 of LNCS, Springer, 2017.
Klein, D., and R. K. Rendsvig, Metrics for Formal Structures, with an Application to Kripke Models and Their Dynamics. arXiv:1704.00977, 2017.
Li, F.-H., C.-T. Li, and M.-K. Shan, Labeled influence maximization in social networks for target marketing, in 2011 IEEE Third Int’l Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third Int’l Conference on Social Computing, 2011, pp. 560–563.
Liu, F., J. Seligman, and P. Girard, Logical dynamics of belief change in the community, Synthese 191(11): 2403–2431, 2014.
Long, C., and R. C.-W. Wong, Minimizing seed set for viral marketing paper, in 2011 IEEE 11th International Conference on Data Mining, 2011, pp. 427–436.
Morris, S., Contagion, Review of Economic Studies 67: 57–78, 2000.
Rendsvig, R. K., Epistemic term-modal logic, in M. Slavkovik, (ed.), Proceedings of the 15th Student Session of The European Summer School in Logic, Language and Information, 2010, pp. 37–46.
Rendsvig, R. K., Diffusion, influence and best-response dynamics in networks: an action model approach, in R. de Haan, (ed.), Proceedings of the ESSLLI 2014 Student Session, Tübingen, 2014, pp. 63–75.
Rendsvig, R. K., Pluralistic ignorance in the bystander effect: informational dynamics of unresponsive witnesses in situations calling for intervention, Synthese 191(11): 2471–2498, 2014.
Rendsvig, R. K., Model transformers for dynamical systems of dynamic epistemic logic, in W. van der Hoek, W. H. Holliday, and W.-F. Wang, (eds.), Logic, Rationality, and Interaction (LORI 2015, Taipei), LNCS, Springer, 2015, pp. 316–327.
Rendsvig, R. K., Logical Dynamics and Dynamical Systems, Ph.D. thesis, Theoretical Philosophy, Lund University, Lund, Sweden, 2018. Lund University Publications.
Ruan, J., and M. Thielscher, A logic for knowledge flow in social networks, in D. Wang, and M. Reynolds, (eds.), AI 2011: Advances in Artificial Intelligence, Vol. 7106 of Lecture Notes in Computer Science, Springer, Berlin, 2011, pp. 511–520.
Schelling, T. C., Models of segregation, The American Economic Review 59(2): 488–493, 1969.
Schelling, T. C., Micromotives and Macrobehavior, Norton, New York, 1978.
Seligman, J., F. Liu, and P. Girard, Logic in the community, in M. Banerjee, and A. Seth, (eds.), Logic and Its Applications, Vol. 6521 of Lecture Notes in Computer Science, Springer, Berlin, 2011, pp. 178–188.
Shakarian, P., S. Eyre, and D. Paulo, A Scalable Heuristic for Viral Marketing Under the Tipping Model. arXiv:1309.2963v1, 2013.
Singh, P., S. Sreenivason, B. K. Szymanski, and G. Korniss, Threshold-limited spreading in social networks with multiple initiators, Scientific Reports 3: 2330, 2013.
Valente, T. W., Social network thresholds in the diffusion of innovations, Social Networks 18(1): 69–89, 1996.
Zhen, L., and J. Seligman, A logical model of the dynamics of peer pressure, Electronic Notes in Theoretical Computer Science 278(0): 275–288, 2011.
We would like to thank the anonymous reviewers for their insightful comments. Zoé Christoff acknowledges support for this research by the Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/M015815/1 “Foundations of Opinion Formation in Autonomous Systems”, as well as from the Deutsche Forschungsgemeinschaft (DFG) and Grantová agentura České republiky (GAČR) joint project RO 4548/6–1 “From Shared Evidence to Group Attitudes”. The contribution of Rasmus K. Rendsvig to the research reported in this article was funded by the Swedish Research Council through the framework project ‘Knowledge in a Digital World’ (Erik J. Olsson, PI). The Center for Information and Bubble Studies is sponsored by the Carlsberg Foundation.
Edited by Giacomo Bonanno, Wiebe Van Der Hoek and Andres Perea
About this article
Cite this article
Baltag, A., Christoff, Z., Rendsvig, R.K. et al. Dynamic Epistemic Logics of Diffusion and Prediction in Social Networks. Stud Logica 107, 489–531 (2019). https://doi.org/10.1007/s11225-018-9804-x
- Social network theory
- Threshold models
- Diffusion in networks
- Social epistemology
- Formal epistemology
- Dynamic epistemic logic
- Opinion dynamics
- Opinion dynamics under uncertainty