Dynamical networks, a.k.a. interacting or coupled dynamical systems on graphs or networks, are a class of spatially extended dynamical systems. On one hand, they aim to elucidate collective dynamics in real-world networks ranging from neuronal networks and swarms of fireflies to communication networks and power grids. On the other hand, they exhibit rich and often unexpected dynamics, which inspire new mathematical research.

The goal of this course is to provide an introduction to dynamical models of networks and the methods of their analysis. We will start with a review of several representative models, including the Kuramoto model of coupled phase oscillators, coupled map lattices, consensus protocols, as well as models for opinion dynamics, alignment, and flocking. In these models, we will identify several common dynamical regimes that are important for understanding collective dynamics. These regimes include synchronization, traveling waves, spatiotemporal chaos, and chimera states. We will then discuss different types of connectivity used in network models, such as all-to-all and nearest-neighbor, as well as Erd\H{o}s-R\'{e}nyi, small-world, and power-law graphs. After that, we will focus on methods for studying dynamical networks, in particular, methods for stability and bifurcation analysis from dynamical systems, elements of algebraic graph theory and the theory of graph limits from discrete mathematics, and continuum/mean-field limit techniques from statistical physics. If time permits, we will also touch upon dynamical systems on fractals and graphs approximating fractals.