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Perspectives on Network Systems and Mathematical Sociology

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

Abstract

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.

Notes

Acknowledgements

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.

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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

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