Abstract
Finding accurate reduced descriptions for large, complex, dynamically evolving networks is a crucial enabler to their simulation, analysis, and ultimately design. Here, we propose and illustrate a systematic and powerful approach to obtaining good collective coarse-grained observables—variables successfully summarizing the detailed state of such networks. Finding such variables can naturally lead to successful reduced dynamic models for the networks. The main premise enabling our approach is the assumption that the behavior of a node in the network depends (after a short initial transient) on the node identity: a set of descriptors that quantify the node properties, whether intrinsic (e.g., parameters in the node evolution equations) or structural (imparted to the node by its connectivity in the particular network structure). The approach creates a natural link with modeling and “computational enabling technology” developed in the context of Uncertainty Quantification. In our case, however, we will not focus on ensembles of different realizations of a problem, each with parameters randomly selected from a distribution. We will instead study many coupled heterogeneous units, each characterized by randomly assigned (heterogeneous) parameter value(s). One could then coin the term Heterogeneity Quantification for this approach, which we illustrate through a model dynamic network consisting of coupled oscillators with one intrinsic heterogeneity (oscillator individual frequency) and one structural heterogeneity (oscillator degree in the undirected network). The computational implementation of the approach, its shortcomings and possible extensions are also discussed.
Highlights
Model reduction for dynamical systems has been an important research direction for decades; accurate reduced models are very useful, and often indispensable for the understanding, analysis, and for the design of large/complex dynamical systems
We can use the coarse timestepper to accelerate the computation of dynamic trajectories of the system, through Coarse Projective Integration (CPI) (Gear and Kevrekidis, 2003; Lee and Gear, 2007)
We have demonstrated that a general network of coupled, intrinsically heterogeneous oscillators can be usefully described using a small number of collective dynamic variables
Summary
Model reduction for dynamical systems has been an important research direction for decades; accurate reduced models are very useful, and often indispensable for the understanding, analysis, and for the design of large/complex dynamical systems. We argue that the same approach which uses the distribution of an intrinsic heterogeneity, and led to the reduction of all-to-all unit assemblies in Moon et al (2006) can be naturally extended to include a distribution over structural heterogeneity that leads to reduction of unit assemblies coupled in networks We demonstrate this in the simplest non-trivial representative setting we can put together: a set of coupled phase oscillators, characterized by heterogeneous frequencies ωi sampled from a prescribed distribution (here a truncated Gaussian)— but not all-to-all coupled. What makes it all possible is the fundamental assumption about how heterogeneity (intrinsic as well as structural) affects the solution: “nearby” parameter values and “nearby” connectivities imply “nearby” dynamics This is not always the case for any network, and so testing that this assumption holds must be performed on a case-by-case basis. Appendices include an analysis of the validity of using higherorder coarse-grained integration schemes
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