Networks provide the natural framework to model the dynamical processes and interactions arising in large-scale multi-agent systems. Whether such interactions are embodied by the exchange of opinions in a social network, data gathering in a sensor network, or signal transduction in a biological network, the topology that determines these interactions plays a key role. This observation has recently led scientists and engineers working in the areas of dynamical processes and control to take a fresh look at networked systems and to develop novel mathematical tools. These efforts are starting to form the basis for a Network Science of dynamical systems. This process is accelerated by the realization that large-scale multi-agent networks (both natural and man-made) are ubiquitous and have a tremendous societal importance. Several researchers at UCSB are active participants in the development of a Network Science of dynamical systems, with extensive expertise in natural large-scale networks (ranging for cellular regulatory networks to ecosystem level networks), in artificial networks (including sensor networks, networks of robots, and the power grid), and in societal networks (such as online social networks and networks of influence and opinion).

A science of large-scale dynamical networks requires fundamentally different approaches and mathematical tools than those traditionally taught in graduate education in dynamical systems and control. This statement is amply supported by the nature of the most recent discoveries in this area, which inevitably resulted from the combination of tools and approaches from a wide range of disciplines. For example, the analysis of consensus networks combines tools from algebraic graph theory, Lyapunov stability, and switched dynamical systems; the design of distributed estimation algorithms uses tools from circuit theory, distributed computation, and vector space optimization; the analysis of routing algorithms for sparse ad-hoc mobile networks can benefit from tools from stochastic process, queuing theory, and partial differential equations; the analysis of synchronization phenomena in power networks relies upon tools from coupled oscillator, statistical physics and non-smooth analysis. The need to combine tools from such a diverse set of areas stems from the need to discover approaches to model, analyze, and control dynamical processes in a manner that scales to a large number of agents and to find innovative ways to bypass combinatorial complexity barriers. In summary, our emphasis is on modern convex optimization methods for the identification of structure and estimation of parameters in large-scale dynamical networks, such as gene expression in biological networks and opinion dynamics in social networks.

**Related Training Modules**

- M1: Spectral Analysis of Dynamic Graphs
- M3: Understanding Scalability of Distributed Estimation Algorithms
- M6: Modeling Network Evolution in Metric Spaces
- M7: Discovery of the Emerging Dynamical Phenomena in Distributed Systems
- M8: Structure and Stability of Bacterial Transcription Networks
- M10: Network Science of Teams
- M17: Models of Social Power Evolution
- M21: Epidemic Propagation Over Contact Networks
- M22: Information Networks in the Moral Narrative Analyzer (MoNA)
- M24: The Value of Information in Zero-Sum Games
- M25: Modeling Gene Network Evolution
- M28: Modeling Structural Balance in Networks
- M29: FLoRa Framework
- M34: Communication Across Networks and Submodular Function Maximization
- M36:
- M39: A Game-theoretic Approach to Mapping Optimality