Systems & Control Letters 57 (2008) 142 – 149 www.elsevier.com/locate/sysconle Area aggregation and time-scale modeling for sparse nonlinear networks  Emrah Bıyık ∗ , Murat Arcak Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA Received 5 July 2006; received in revised form 2 August 2007; accepted 21 August 2007 Available online 31 October 2007 Abstract Model reduction and aggregation are of key importance for simulation and analysis of large-scale systems, such as molecular dynamics, large swarms of robotic vehicles, and animal aggregations. We study a nonlinear network which exhibits areas of internally dense and externally sparse interconnections. The densely connected nodes in these areas synchronize in the fast time-scale, and behave as aggregate nodes that dominate the slow dynamics of the network. We first derive a singular perturbation model which makes this time-scale separation explicit and, next, prove the validity of the reduced-model approximation on the infinite time interval. © 2007 Elsevier B.V. All rights reserved. Keywords: Large-scale systems; Sparse networks; Area aggregation; Singular perturbation; Cooperative control 1. Introduction One of the most significant problems in large-scale systems research is the prediction of the dynamic behavior for the full network from a computationally tractable reduced-order model. A powerful approach for the development of such reduced mod- els is to exploit the separation of time scales that arise due to differing strengths in the connections between the components. A time-scale decomposition methodology was developed in the 1970s for large power systems, and reduced network models for long-term behavior were obtained by representing each co- herent area with a single “equivalent machine” [3]. Analogous results exist in large economic models where, in the long run, each sub-economy is represented by a single aggregate price, which is then used to study the interaction between these sub- economies [13]. In this paper we show that a time-scale separation also re- sults from a sparse interconnection structure where coherent areas are now characterized by internally dense and externally sparse interconnections. Sparse interconnection structures are  This work was supported in part by NSF under award no. ECS-0238268 and Air Force Office of Scientific Research under award no. FA9550-07-1- 0308. ∗ Corresponding author. E-mail addresses: biyike@rpi.edu (E. Bıyık), arcakm@rpi.edu (M. Arcak). 0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.08.003 ubiquitous in numerous natural and technological networks such as cell cycle regulatory networks [9], small-world net- works [15,17], social networks [16], sensor networks [18], and in hierarchical routing in the Internet [7]. In [5], Chow and Kokotovi´ c studied a class of linear network models that exhibit such a sparsity structure, and developed a singular perturbation decomposition. They further studied power system models where densely connected generators in the same area syn- chronize in the fast-time scale, and constitute aggregate nodes which characterize the long-term behavior of the system. Our result in this paper is applicable to nonlinear networks and encompasses as a special case the linear study of [5]. It further employs modern tools from algebraic graph theory and pro- vides new insights on the structure of the singular perturbation decomposition, even for the linear case. The nonlinear result is made possible by a new set of slow and fast variables moti- vated by the graph Laplacian, and by a recent singular pertur- bation approximation result in the infinite time interval due to Khalil [8]. The subsequent sections are organized as follows. In Sec- tion 2, the dynamic network model and the agreement problem is introduced. Section 3 motivates our area aggregation and time-scale separation study on a formation control example. Section 4 characterizes the sparse interconnection structure of the network. Section 5 introduces slow and fast variables that reveal the time-scale separation in the network. Section 6