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Mathematical Tools and Convergence Results for Dynamics over Networks

Abstract

Mathematical models of networked dynamical systems are ubiquitous - they are used to study power grids, networks of webpages, robotic and sensor networks, and social networks, to name a few. Importantly, most real-world networks are time-varying and are affected by stochastic phenomena such as adversarial attacks and communication link failures. Time-varying networks, therefore, have been under study for a few decades. However, our current understanding of the dynamical processes evolving over such networks is limited. This observation motivates the two-pronged objective of this dissertation: (i) to use theoretical and empirical methods to analyze certain networked dynamical systems that cannot be studied using standard tools and techniques, and (ii) to develop suitable mathematical techniques for the systematic study of such systems.

As our main contribution resulting from (i), we use the properties of random time-varying networks to provide a rigorous theoretical foundation for the age-structured Susceptible-Infected-Recovered model, a model of epidemic spreading. We then use system identification to show that the age-structured SIR dynamics accurately model the spread of COVID-19 at city and state levels in two different parts of the world – Tokyo and California.

As for our contributions resulting from (ii), we extend two assertions of the Perron-Frobenius theorem to time-varying networks described by strongly aperiodic stochastic chains, thereby widening the applicability of the fundamental result that is foundational to probability theory and to the studies of complex networks, population dynamics, internet search engines, etc. Our results enable us to extend several known results on distributed learning and averaging. Moreover, they promise to advance our understanding of dynamical processes over real-world networks.

As an application of these results, we study non-Bayesian social learning on random time-varying networks that violate the standard connectivity criterion of uniform strong connectivity. In doing so, we also make a methodological contribution: we show how the theory of absolute probability sequences and martingale theory can be combined to analyze nonlinear dynamics that approximate linear dynamics asymptotically in time.

Finally, we study the convergence properties of social Hegselmann-Krause dynamics (which is a variant of the classical Hegselmann-Krause model of opinion dynamics and incorporates state-dependence into distributed averaging). As our main contribution here, we provide nearly tight necessary and sufficient conditions for a given connectivity graph to exhibit unbounded epsilon-convergence times for such dynamics.

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