Introduction to the Modeling and Analysis of Complex Systems

Chapter 18Dynamical Networks III: Analysis ofNetwork Dynamics18.1 Dynamics of Continuous-State NetworksWe will now switch gears to the analysis of dynamical properties of networks. We willfirst discuss how some of the analytical techniques we already covered in earlier chapterscan be applied to dynamical network models, and then we will move onto some additionaltopics that are specific to networks.First of all, I would like to make it clear that we were already discussing dynamicalnetwork models in earlier chapters. A typical autonomous discrete-time dynamical systemx t = F (x t−1 ), (18.1)or a continuous-time onedxdt= F (x), (18.2)can be considered a dynamical network if the state space is multidimensional. For example,a system with a five-dimensional state space can be viewed as a dynamical networkmade of five nodes, each having a scalar state that changes dynamically based on themathematical rule determined in function F (Fig. 18.1). More specifically, the dynamics ofnode i’s state is determined by the i-th dimensional part of F , and if that part refers to thestate vector’s j-th component, then node j is connected to node i, and so on.This means that dynamical networks are not fundamentally different from other dynamicalsystems. Therefore, if the node states are continuous, then all the analytical405