UC Santa Barbara

Networks are ubiquitous in nature and engineering, with applications in areas such as modeling power grids, population dynamics, and social networks. Applying network science to complex dynamical systems provides us with powerful tools to study the interactions between agents. Moreover, these tools can also be used to study the nonlinear dynamics of the network itself and he combination of dynamics over networks and dynamics of networks paves the way for new models that accurately capture the rich behavior in sociological processes. In this thesis, we first study synchronization in networks of coupled oscillators. Second we model and analyze the dynamics of influence networks of human teams.

Regarding coupled oscillators, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature.

Regarding the dynamics of social networks of human teams, we focus on modeling and analyzing how performance and expertise impact the level of influence team members have on each other. First, we propose a novel quantitative model describing the decentralized process by which individuals in a team learn who has what abilities, while concurrently assigning tasks to each of the team members. Our theoretical analysis characterizes a team's ability, or inability, to learn each other's skill and thus converge to a work allocation maximizing the team performance. Second, we propose a cognitive dynamical model to describe the process by interpersonal influences are adjusted in small teams over a sequence of intellective tasks with fixed workload. We provide analytical results on the model's asymptotic behavior for the case with identically performing individuals and verify the accuracy of the proposed model on experimental data.