Mason A. Porter, James P. Gleeson.
Dynamical Systems on Networks: a tutorial.
Springer. 2016
rating : 3.5 : worth reading
review : 9 June 2024
This tutorial, which also includes key pointers to the literature, should be helpful for junior and senior undergraduate students, graduate students, and researchers from mathematics, physics, and engineering who seek to study dynamical systems on networks but who may not have prior experience with graph theory or networks.
This slim volume (80pp) provides a condensed overview of some aspects of dynamical systems on networks.
A dynamical system is one that explains how its state changes with time, either continuous time (typically using an ordinary differential equation, or ODE), or discrete time (typically using a difference equation, also called a map). When multiple similar states are coupled, so that one state is influenced by its own and its neighbours’ states, this is dynamics on a network, where the network describes the topology of the coupling. For example, the network might capture a collection of oscillators coupled in a line, or a collection of contagious individuals or populations coupled through a social or proximity network.
This tutorial focusses on analytic solutions, describing some of the approaches to simplification and approximation necessary for tractability. It includes examples of percolation, bio-contagion (illness), social contagion (memes), voter models, coupled oscillators, and a variety of other processes. It concludes with two very brief chapters on software implementations (2pp) and dynamics of networks (where the structure of the network itself is dynamic; 3pp).
The brevity makes for a quick read, lightly covering a lot of material, but one is left wishing for much more. One of the most useful parts is the extensive bibliography of 335 references. Newman’s much more substantial book on Networks is heavily cited throughout; maybe this should be my next stop?