Network Sparsification with Guaranteed Systemic Performance Measures

Abstract

A proper abstraction of a large-scale linear consensus network with a dense coupling graph is one whose number of coupling links is proportional to its number of subsystems and its performance is comparable to the original network. Optimal design problems for an abstracted network are more amenable to efficient optimization algorithms. From the implementation point of view, maintaining such networks are usually more favorable and cost effective due to their reduced communication requirements across a network. Therefore, approximating a given dense linear consensus network by a suitable abstract network is an important analysis and synthesis problem. In this paper, we develop a framework to compute an abstraction of a given large-scale linear consensus network with guaranteed performance bounds using a nearlylinear time algorithm. First, the existence of abstractions of a given network is proven. Then, we present an efficient and fast algorithm for computing a proper abstraction of a given network. Finally, we illustrate the effectiveness of our theoretical findings via several numerical simulations. † M. Siami is with the Institute for Data, Systems, and Society,

Growing Linear Dynamical Networks Endowed by Spectral Systemic Performance Measures

“…This can done by relaxing and replacing the orthogonal invariance property by permutation invariance property. The local deviation error is an example of a non-spectral systemic performance measure [18], [37]. Our ongoing research involves a comprehensive treatment of this class of measures.…”

Optimal network design for synchronization of coupled oscillators

“…The framework herein proposed can be used in a wide range of practical applications, namely, prevention of cascading failures in power grids (Linnemann et al, 2011), optimal design of electrical infrastructure upgrades, sparsity promoting network design (Siami and Motee, 2015;Dhingra et al, 2012;Lin et al, 2012), and detection of links inducing the Braess' paradox (i.e., the counter-intuitive phenomenon of losing synchronization as the result of adding new edges (Witthaut and Timme, 2012)). We will discuss some of these applications in §5.…”

Time-Delay Origins of Fundamental Tradeoffs Between Risk of Large Fluctuations and Network Connectivity

“…The focus of these works are on linear consensus networks with no time delay. In [38,35], the authors introduce a class of spectral systemic performance measures (which includes H p -norms as its special case) to synthesize networks through growing and sparsification of links in noisy linear consensus networks in absence of time delay. In all these papers, performance measures assess holistic and macroscopic features of the networks, such as total deviation from average or uncertainty volume of the output.…”