The Evolution of Beliefs over Signed Social Networks

Guodong Shi (), Alexandre Proutiere (), Mikael Johansson (), John S. Baras () and Karl H. Johansson ()
Additional contact information
Guodong Shi: Research School of Engineering, CECS, The Australian National University, Canberra ACT 0200, Australia
Alexandre Proutiere: ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 10044, Sweden
Mikael Johansson: ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 10044, Sweden
John S. Baras: Institute for Systems Research, University of Maryland, College Park, Maryland 20742
Karl H. Johansson: ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 10044, Sweden

Operations Research, 2016, vol. 64, issue 3, 585-604

Abstract: We study the evolution of opinions (or beliefs) over a social network modeled as a signed graph. The sign attached to an edge in this graph characterizes whether the corresponding individuals or end nodes are friends (positive links) or enemies (negative links). Pairs of nodes are randomly selected to interact over time, and when two nodes interact, each of them updates its opinion based on the opinion of the other node and the sign of the corresponding link. This model generalizes the DeGroot model to account for negative links: when two adversaries interact, their opinions go in opposite directions. We provide conditions for convergence and divergence in expectation, in mean-square, and in almost sure sense and exhibit phase transition phenomena for these notions of convergence depending on the parameters of the opinion update model and on the structure of the underlying graph. We establish a no-survivor theorem, stating that the difference in opinions of any two nodes diverges whenever opinions in the network diverge as a whole. We also prove a live-or-die lemma, indicating that almost surely, the opinions either converge to an agreement or diverge. Finally, we extend our analysis to cases where opinions have hard lower and upper limits. In these cases, we study when and how opinions may become asymptotically clustered to the belief boundaries and highlight the crucial influence of (strong or weak) structural balance of the underlying network on this clustering phenomenon.

Keywords: opinion dynamics; signed graph; social networks; opinion clustering (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (18)

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