Coherence and Concentration in Tightly-Connected Networks

Abstract

The ability to achieve coordinated behaviorengineered or emergent-on networked systems has attracted widespread interest over several fields. This interest has led to remarkable advances in developing a theoretical understanding of the conditions under which agents within a network can reach an agreement (consensus) or develop coordinated behavior, such as synchronization. However, much less understood is the phenomenon of network coherence. Network coherence generally refers to nodes' ability in a network to have a similar dynamic response despite heterogeneity in their individual behavior. In this paper, we develop a general framework to analyze and quantify the level of network coherence that a system exhibits by relating coherence with a low-rank property of the system. More precisely, for a networked system with linear dynamics and coupling, we show that, as the network connectivity grows, the system transfer matrix converges to a rank-one transfer matrix representing the coherent behavior. Interestingly, the non-zero eigenvalue of such a rank-one matrix is given by the harmonic mean of individual nodal dynamics, and we refer to it as the coherent dynamics. Our analysis unveils the frequency-dependent nature of coherence and a non-trivial interplay between dynamics and network topology. We further show that many networked systems can exhibit similar coherent behavior by establishing a concentration result in a setting with randomly chosen individual nodal dynamics.

Spectral clustering and model reduction for weakly-connected coherent network systems

“…Recent work [11], [12] has shown that, under mild assumptions, the following holds 1 for almost any s 0 ∈ C,…”

“…Recently, it has been theoretically established that coherence naturally emerges when a group of nodes are tightlyconnected, regardless of the node dynamics, as long as the interconnection remains stable [11], [12]. The analysis also provides an asymptotically (as the network connectivity increases) exact characterization of the coherent response, H. Min and E. Mallada are with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA{hanchmin, mallada}@jhu.edu.…”

“…Intuitively, when we apply those clustering algorithms to identify tightly-connected components in the network, each component should be coherent also in the dynamical sense. Then, applying [11], [12] for each cluster, should lead to a good model for each coherent group, which after interconnected with an appropriately chosen reduced graph should lead to a good networkreduced aggregate model of the dynamic interactions among across coherent components.…”

“…In this paper, we formalize and theoretically justify this seemingly naïve approach utilizing the recent frequencydomain analysis for coherence [12] and dynamics aggregation [14]. Specifically, we prose a novel approximation model for large-scale networks with two tightly-connected components/groups.…”

“…bus phase angles. Finally, in transportation networks[16], g i (s) represent the vehicle dynamics whereas f (s)L describes local inter-vehicle information transfer.Recent work[11],[12] has shown that, under mild assumptions, the following holds 1 for almost any s 0 ∈ C,…”