I will present a model of continuous-time opinion dynamics for an arbitrary number of agents that communicate over a network and form real-valued opinions about an arbitrary number of options.  The model generalizes linear and nonlinear models in the literature. Drawing from biology, physics, and social psychology, we introduce an attention parameter to modulate social influence and a saturation function to bound inter-agent and intra-agent opinion exchanges.  This yields simply parameterized dynamics that exhibit the range of opinion formation behaviors predicted by model-independent bifurcation theory but not exhibited by linear models or existing nonlinear models. Behaviors include reliable formation of consensus and dissensus, even in homogeneous networks, and opinion cascades. The opinion dynamics also display ultra-sensitivity to inputs, robustness to disturbance, and flexible transitions between consensus and dissensus. Augmenting the opinion dynamics with feedback dynamics for the attention parameter results in tunable thresholds that govern sensitivity, robustness, and flexibility.  The model provides new means for systematic study of dynamics on natural and engineered networks, from information spread and political polarization to collective decision making and dynamic task allocation. This is joint work with Alessio Franci (UNAM, Mexico) and Anastasia Bizyaeva (Princeton).

Naomi Ehrich Leonard is a control theorist whose work involves analysis and design of feedback and interconnection in complex, dynamical systems.  She uses mathematical models and methods to study mechanisms of collective motion and collective decision making for multi-agent systems in nature (analysis of animal and human groups) and in engineering (design of autonomous robotic teams and mobile sensor networks).  She has applied her work to the collective dynamics of killifish, starlings, honeybees, zebras, and desert harvester ants, as well as to rule-based improvisational dance.  She led a multidisciplinary ocean sensing project with a month-long deployment of an automated, adaptive network of underwater robotic vehicles in Monterey Bay, CA.  Leonard is the Edwin S. Wilsey Professor of Mechanical and Aerospace Engineering and associated faculty member of the Program in Applied and Computational Mathematics at Princeton University.  She is a MacArthur Fellow, a Fellow of the American Academy of Arts and Sciences, IEEE, SIAM, ASME, and IFAC. Visit her website here.

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