Sparsity and Time Scales

This paper develops an asymptotic approach to the analysis of dynamic networks with dense and sparse connections. A measure of the sparsity is shown to have a meaning similar to the weak connection parameter in the slow coherency method. The asymptotic sparsity analysis reveals the effect of the system dimension on the slow aggregate model and provides an interpretation for the inclusion of a higher order term neglected in the weak connection analysis. Two network sequences with different types of sparsity are used to demonstrate how the sparsity properties influence the dynamic behavior and lead to a separation of time scales.