Coarse-graining dynamical networks

Abstract

Network-based modeling of complex systems with emergent behavior is increasingly being employed across different disciplines. Much of the focus in such efforts has been on detailed, "microscopic" modeling, in which the dynamics is described in terms of the individual nodes and edges of the network. When the network sizes become very large, however, model reduction approaches become crucial in helping us understand the interplay between the structure and the dynamics of networks. The primary goal of this dissertation is to develop and computationally implement coarse-graining approaches that can significantly enhance our understanding of macroscopic network dynamics.

Computer-assisted reduced models of dynamical networks are presented through three illustrative problems by using a model reduction framework known as the equation-free approach. This technique relies on the knowledge of "good" coarse observables, and suitably defined operations to convert between the detailed description and the coarse description. The basic idea is to estimate the information necessary for coarse-grained computations on the fly, using short bursts of appropriately designed detailed simulations. This computational framework facilitates efficient computation of models by accelerating simulations and also by enabling additional modeling tasks, such as coarse fixed point/bifurcation/stability computations, even when an explicit coarse-grained model is not available.

The three illustrative models discussed in this dissertation are: (1) coupled oscillators connected by a network structure, (2) information propagation on a social network, and (3) a random evolution of networks. However, the methods presented and implemented in this work are quite general and can be used with any suitable dynamic network problem with an appropriate choice of reduced observables.