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Finding accurate reduced descriptions for large, complex, dynamicallyevolving networks is a crucial enabler to their simulation, analysis, and,ultimately, design. Here we propose and illustrate a systematic and powerfulapproach to obtaining good collective coarse-grained observables-- variablessuccessfully summarizing the detailed state of such networks. Finding suchvariables can naturally lead to successful reduced dynamic models for thenetworks. The main premise enabling our approach is the assumption that thebehavior of a node in the network depends (after a short initial transient) onthe node identity: a set of descriptors that quantify the node properties,whether intrinsic (e.g. parameters in the node evolution equations) orstructural (imparted to the node by its connectivity in the particular networkstructure). The approach creates a natural link with modeling and"computational enabling technology" developed in the context of UncertaintyQuantification. In our case, however, we will not focus on ensembles ofdifferent realizations of a problem, each with parameters randomly selectedfrom a distribution. We will instead study many coupled heterogeneous units,each characterized by randomly assigned (heterogeneous) parameter value(s). Onecould then coin the term Heterogeneity Quantification for this approach, whichwe illustrate through a model dynamic network consisting of coupled oscillatorswith one intrinsic heterogeneity (oscillator individual frequency) and onestructural heterogeneity (oscillator degree in the undirected network). Thecomputational implementation of the approach, its shortcomings and possibleextensions are also discussed.

Coarse-Grained Descriptions of Dynamics for Networks with both Intrinsic and Structural Heterogeneities