Analysis and Design of Robust and High-Performance Complex Dynamical Networks

In the first part of this dissertation, we develop some basic principles to investigate performance deterioration of dynamical networks subject to external disturbances. First, we propose a graph-theoretic methodology to relate structural specifications of the coupling graph of a linear consensus network to its performance measure. Moreover, for this class of linear consensus networks, we introduce new insights into the network centrality based not only on the network graph but also on a more structured model of network uncertainties. Then, for the class of generic linear networks, we show that the H_2-norm, as a performance measure, can be tightly bounded from below and above by some spectral functions of state and output matrices of the system. Finally, we study nonlinear autocatalytic networks and exploit their structural properties to characterize their existing hard limits and essential tradeoffs. In the second part, we consider problems of network synthesis for performance enhancement. First, we propose an axiomatic approach for the design and performance analysis of linear consensus networks by introducing a notion of systemic performance measure. We build upon this new notion and investigate a general form of combinatorial problem of growing a linear consensus network via minimizing a given systemic performance measure. Two efficient polynomial-time approximation algorithms are devised to tackle this network synthesis problem. Then, we investigate the optimal design problem of distributed system throttlers. A throttler is a mechanism that limits the flow rate of incoming metrics, e.g., byte per second, network bandwidth usage, capacity, traffic, etc. Finally, a framework is developed to produce a sparse approximation of a given large-scale network with guaranteed performance bounds using a nearly-linear time algorithm

Disturbance Propagation in Interconnected Linear Dynamical Networks

We consider performance analysis of interconnected linear dynamical networks subject to external stochastic disturbances. For stable linear networks, we define scalar performance measures by considering weighted H2--norms of the underlying systems, which are defined from the disturbance input to a desired output. It is shown that the performance measure of a general stable linear network can be tightly bounded from above and below using some spectral functions of the state matrix of the network. This result is applied to a class of cyclic linear networks and shown that the performance measure of such networks scales quadratically with the network size. Next, we focus on first-- and second--order linear consensus networks and introduce the notion of Laplacian energy for such networks, which in fact measures the expected steady-state dispersion of the state of the entire network. We develop a graph-theoretic framework in order to relate graph characteristics to the Laplacian energy of the network and show that how the Laplacian energy scales asymptotically with the network size. We quantify several inherent fundamental limits on Laplacian energy in terms of graph diameter, node degrees, and the number of spanning trees, and several other graph specifications. Particularly we characterize several versions of fundamental tradeoffs between Laplacian energy and sparsity measures of a linear consensus network, showing that more sparse networks have higher levels of Laplacian energies. At the end, we show that several existing performance measures in real--world applications, such as total power loss in synchronous power networks and flock energy of a group of autonomous vehicles in a formation, are indeed special forms of Laplacian energies