Perspectives on Network Systems and Mathematical Sociology

  • Francesco BulloEmail author
  • Noah E. Friedkin
Part of the Systems & Control: Foundations & Applications book series (SCFA)


This chapter reviews selected topics in network systems and mathematical sociology. We first review classic results on Perron–Frobenius and algebraic graph theory. We then focus on mathematical sociology and describe models of opinion dynamics in social influence systems, including the classic French–Harary–DeGroot and the Friedkin–Johnsen models. Based on recent controlled experiments, we present recent empirical results on opinion dynamics along single issues and sequences of issues. Finally, motivated by these empirical results, we describe some mathematical models for the evolution of social power and influence systems via the reflected appraisal mechanism.



This chapter is humbly prepared in honor of our dear friend and collaborator Roberto Tempo. Both authors are deeply thankful for the opportunity to collaborate with him on the themes of this article during the last years of his career. His visits to Santa Barbara were full with intellectual excitement, unending discussions, friendship, and joy. His intelligence, humor, positive and friendly attitude, and wisdom are already sorely missed.

This material is based upon work supported by, or in part by, the U.S. ARL and the U.S. ARO under grant W911NF-15-1-0577, the AFOSR under grant FA9550-15-1-0138, and the DOE under grant XAT-6-62531-01. The first author thanks Florian Dörfler and Alessandro Giua for insightful conversations on the topics of this document.

The first part of this chapter is a highly abbreviated version of the first part of the textbook [7]. All images in this chapter are reproduced here with permission from [7] – they are licensed under the Creative Commons BY-SA 4.0 Attribution-ShareAlike International License. Tables 1 and 2 are reproduced here with permission from [24] – they are licensed under the Creative Commons SA 4.0 Attribution International License.


  1. 1.
    R. P. Abelson. Mathematical models of the distribution of attitudes under controversy. In N. Frederiksen and H. Gulliksen, editors, Contributions to Mathematical Psychology, volume 14, pages 142–160. Holt, Rinehart, & Winston, 1964. ISBN 0030430100.Google Scholar
  2. 2.
    M. Arcak, C. Meissen, and A. Packard. Networks of Dissipative Systems: Compositional Certification of Stability, Performance, and Safety. Springer, 2016. ISBN 978-3-319-29928-0. Scholar
  3. 3.
    H. Bai, M. Arcak, and J. Wen. Cooperative Control Design. Springer, 2011. ISBN 1461429072.Google Scholar
  4. 4.
    D. Bindel, J. Kleinberg, and S. Oren. How bad is forming your own opinion? Games and Economic Behavior, 92: 248–265, 2015. Scholar
  5. 5.
    G. Birkhoff. Extensions of Jentzsch’s theorem. Transactions of the American Mathematical Society, 85 (1): 219–227, 1957. Scholar
  6. 6.
    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang. Complex networks: Structure and dynamics. Physics Reports, 424 (4-5): 175–308, 2006. Scholar
  7. 7.
    F. Bullo. Lectures on Network Systems. CreateSpace, 1 edition, 2018. ISBN 978-1986425643. With contributions by J. Cortés, F. Dörfler, and S. Martínez.
  8. 8.
    F. Bullo, J. Cortés, and S. Martínez. Distributed Control of Robotic Networks. Princeton University Press, 2009. ISBN 978-0-691-14195-4. URL
  9. 9.
    Y. Cao, W. Yu, W. Ren, and G. Chen. An overview of recent progress in the study of distributed multi-agent coordination. IEEE Transactions on Industrial informatics, 9 (1): 427–438, 2013. Scholar
  10. 10.
    C. Castellano, S. Fortunato, and V. Loreto. Statistical physics of social dynamics. Reviews of Modern Physics, 81 (2): 591–646, 2009. Scholar
  11. 11.
    G. Chen, X. Duan, N. E. Friedkin, and F. Bullo. Social power dynamics over switching and stochastic influence networks. IEEE Transactions on Automatic Control, 2018. To appear.
  12. 12.
    X. Chen, J. Liu, M.-A. Belabbas, Z. Xu, and T. Başar. Distributed evaluation and convergence of self-appraisals in social networks. IEEE Transactions on Automatic Control, 62 (1): 291–304, 2017. Scholar
  13. 13.
    C. H. Cooley. Human Nature and the Social Order. Charles Scribner Sons, New York, 1902.Google Scholar
  14. 14.
    E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Pedestrian Dynamics. Springer, 2014. ISBN 978-3-319-06619-6.Google Scholar
  15. 15.
    M. H. DeGroot. Reaching a consensus. Journal of the American Statistical Association, 69 (345): 118–121, 1974. Scholar
  16. 16.
    D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, 2010. ISBN 0521195330.Google Scholar
  17. 17.
    B. A. Francis and M. Maggiore. Flocking and Rendezvous in Distributed Robotics. Springer, 2016. ISBN 978-3-319-24727-4.Google Scholar
  18. 18.
    P. Frasca, H. Ishii, C. Ravazzi, and R. Tempo. Distributed randomized algorithms for opinion formation, centrality computation and power systems estimation: A tutorial overview. European Jounal of Control, 24: 2–13, 2015. Scholar
  19. 19.
    J. R. P. French Jr. A formal theory of social power. Psychological Review, 63 (3): 181–194, 1956. Scholar
  20. 20.
    N. E. Friedkin. A formal theory of reflected appraisals in the evolution of power. Administrative Science Quarterly, 56 (4): 501–529, 2011. Scholar
  21. 21.
    N. E. Friedkin and F. Bullo. How truth wins in opinion dynamics along issue sequences. Proceedings of the National Academy of Sciences, 114 (43): 11380–11385, 2017. Scholar
  22. 22.
    N. E. Friedkin and E. C. Johnsen. Social influence networks and opinion change. In S. R. Thye, E. J. Lawler, M. W. Macy, and H. A. Walker, editors, Advances in Group Processes, volume 16, pages 1–29. Emerald Group Publishing Limited, 1999. ISBN 0762304529.Google Scholar
  23. 23.
    N. E. Friedkin and E. C. Johnsen. Social Influence Network Theory: A Sociological Examination of Small Group Dynamics. Cambridge University Press, 2011. ISBN 9781107002463.Google Scholar
  24. 24.
    N. E. Friedkin, P. Jia, and F. Bullo. A theory of the evolution of social power: Natural trajectories of interpersonal influence systems along issue sequences. Sociological Science, 3: 444–472, 2016a. Scholar
  25. 25.
    N. E. Friedkin, A. V. Proskurnikov, R. Tempo, and S. E. Parsegov. Network science on belief system dynamics under logic constraints. Science, 354 (6310): 321–326, 2016b. Scholar
  26. 26.
    F. R. Gantmacher. The Theory of Matrices, volume 1 and 2. Chelsea, New York, 1959. ISBN 0-8218-1376-5 and 0-8218-2664-6. Translation of German edition by K. A. Hirsch.Google Scholar
  27. 27.
    F. Garin and L. Schenato. A survey on distributed estimation and control applications using linear consensus algorithms. In A. Bemporad, M. Heemels, and M. Johansson, editors, Networked Control Systems, LNCIS, pages 75–107. Springer, 2010. Scholar
  28. 28.
    V. Gecas and M. L. Schwalbe. Beyond the looking-glass self: Social structure and efficacy-based self-esteem. Social Psychology Quarterly, 46 (2): 77–88, 1983. Scholar
  29. 29.
    F. Harary. A criterion for unanimity in French’s theory of social power. In D. Cartwright, editor, Studies in Social Power, pages 168–182. University of Michigan, 1959. ISBN 0879442301.
  30. 30.
    R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1985. ISBN 0521386322.Google Scholar
  31. 31.
    M. O. Jackson. Social and Economic Networks. Princeton University Press, 2010. ISBN 0691148201.Google Scholar
  32. 32.
    P. Jia, N. E. Friedkin, and F. Bullo. Opinion dynamics and social power evolution: A single-timescale model. IEEE Transactions on Control of Network Systems, Dec. 2017a. Submitted.Google Scholar
  33. 33.
    P. Jia, N. E. Friedkin, and F. Bullo. Opinion dynamics and social power evolution over reducible influence networks. SIAM Journal on Control and Optimization, 55 (2): 1280–1301, 2017b. Scholar
  34. 34.
    P. Jia, A. MirTabatabaei, N. E. Friedkin, and F. Bullo. Opinion dynamics and the evolution of social power in influence networks. SIAM Review, 57 (3): 367–397, 2015. Scholar
  35. 35.
    S. Martínez, J. Cortés, and F. Bullo. Motion coordination with distributed information. IEEE Control Systems, 27 (4): 75–88, 2007.
  36. 36.
    M. Mesbahi and M. Egerstedt. Graph Theoretic Methods in Multiagent Networks. Princeton University Press, 2010. ISBN 9781400835355.Google Scholar
  37. 37.
    A. MirTabatabaei, P. Jia, N. E. Friedkin, and F. Bullo. On the reflected appraisals dynamics of influence networks with stubborn agents. In American Control Conference, pages 3978–3983, Portland, OR, USA, June 2014.
  38. 38.
    M. E. J. Newman. The structure and function of complex networks. SIAM Review, 45 (2): 167–256, 2003. Scholar
  39. 39.
    M. E. J. Newman. Networks: An Introduction. Oxford University Press, 2010. ISBN 0199206651.Google Scholar
  40. 40.
    K.-K. Oh, M.-C. Park, and H.-S. Ahn. A survey of multi-agent formation control. Automatica, 53: 424–440, 2015. Scholar
  41. 41.
    S. E. Parsegov, A. V. Proskurnikov, R. Tempo, and N. E. Friedkin. A new model of opinion dynamics for social actors with multiple interdependent attitudes and prejudices. In IEEE Conf. on Decision and Control, pages 3475–3480, 2015.Google Scholar
  42. 42.
    S. E. Parsegov, A. V. Proskurnikov, R. Tempo, and N. E. Friedkin. Novel multidimensional models of opinion dynamics in social networks. IEEE Transactions on Automatic Control, 62 (5): 2270–2285, 2017. Scholar
  43. 43.
    A. V. Proskurnikov and R. Tempo. A tutorial on modeling and analysis of dynamic social networks. Part I. Annual Reviews in Control, 43: 65–79, 2017. Scholar
  44. 44.
    C. Ravazzi, P. Frasca, R. Tempo, and H. Ishii. Ergodic randomized algorithms and dynamics over networks. IEEE Transactions on Control of Network Systems, 2 (1): 78–87, 2015. Scholar
  45. 45.
    W. Ren and R. W. Beard. Distributed Consensus in Multi-vehicle Cooperative Control. Communications and Control Engineering. Springer, 2008. ISBN 978-1-84800-014-8.zbMATHCrossRefGoogle Scholar
  46. 46.
    W. Ren, R. W. Beard, and E. M. Atkins. Information consensus in multivehicle cooperative control. IEEE Control Systems, 27 (2): 71–82, 2007.
  47. 47.
    E. Seneta. Non-negative Matrices and Markov Chains. Springer, 2 edition, 1981. ISBN 0387297650.Google Scholar
  48. 48.
    R. Sepulchre, A. Sarlette, and P. Rouchon. Consensus in non-commutative spaces. In IEEE Conf. on Decision and Control, pages 6596–6601, Atlanta, USA, Dec. 2010.
  49. 49.
    H. A. Simon. Administrative Behavior. A Study of Decision-making Processes in Administrative Organization. Free Press, 1947.Google Scholar
  50. 50.
    J. N. Tsitsiklis, D. P. Bertsekas, and M. Athans. Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Transactions on Automatic Control, 31 (9): 803–812, 1986. Scholar
  51. 51.
    M. Ye, J. Liu, B. D. O. Anderson, C. Yu, and T. Başar. Evolution of social power in social networks with dynamic topology. IEEE Transactions on Automatic Control, 2018. To appear.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering and Center for Control, Dynamical-Systems and Computation, 2325 Engineering Bldg IIUniversity of California at Santa BarbaraSanta BarbaraUSA
  2. 2.Department of Sociology and Center for Control, Dynamical-Systems and ComputationUniversity of California at Santa BarbaraSanta BarbaraUSA

Personalised recommendations