Kutz, Jose NMorrison, Megan Jean2021-10-292021-10-292021Morrison_washington_0250E_22783.pdfhttp://hdl.handle.net/1773/47917Thesis (Ph.D.)--University of Washington, 2021Networks in nature regularly exhibit dynamics that are difficult to characterize due to their nonlinear nature and use of obscure control signals. These systems are often marked by low-dimensional dynamics, multiple stable fixed points or attractors, and outputs that are generated from nonlocalized network activity. We develop procedures for characterizing nonlinear dynamics in networks with transparent, low-dimensional models and controlling them using bifurcation theory. We apply these techniques to the neural network of C. elegans, Hopfield networks, and randomly generated high-dimensional dynamical systems. We show that nonlinear control may be a method by which C. elegans regulates its behavior and could be a viable control method in other systems with multiple stable fixed points. Although much focus rests on the dynamics of nodes in a network, many networks of interest, such as sociopolitical networks, possess edge dynamics in addition to node dynamics. One such network is the international system. We explore dimension reduction techniques and the governing equations for network edge dynamics in addition to node dynamics. We show how final stable states can be predicted from initial network statistics for random matrices under the influence of structural balance dynamics; this analysis is useful for understanding when assortativity in a network, which can occur due to in-group biases, will determine the factionalization that occurs in networks under structural balance dynamics. We further build a matrix dynamical systems model of sociopolitical edge dynamics with low-conflict and high-conflict stable states. We analyze the edge dynamics in a low-dimensional eigenvalue/eigenvector space and derive bifurcations for state transitions in the eigenvalue space; this is similar to our derivation of state transitions for node dynamics. Used together, data-driven discovery, dimension reduction, and bifurcation theory can be used to effectively describe, analyze, and control network dynamics.Data-driven techniques allow us to identify the dynamics governing activity in complex networks. Dimension reduction allows us to characterize high-dimensional dynamics with far fewer variables. Bifurcation theory allows us to understand how and why qualitative transitions occur in nonlinear systems. We demonstrate these techniques on several systems including toy models, random networks, the nematode C. elegans neural network, and the European international system. We hope that these strategies for building models for network dynamics and evaluating control techniques can be useful in a wider range of networks with nonlinear dynamics.application/pdfen-USCC BY-NC-NDbifurcation theorydimension reductiondynamical systemsnetwork sciencesocial sciencestheoretical neuroscienceMathematicsApplied mathematicsThesis