A Structurally-Based Approach to Nonlinear Analysis and Design of Networks
National Science Foundation EECS 0852750
The dynamic behavior of various natural and technological networks is dictated by the structure of interactions between the components rather than by detailed properties of the components. This project is developing a new approach that exploits the network structure while broadly characterizing the components by their essential properties, such as passivity and time-scales, that are relevant to the ensemble behavior. This structurally-based approach promises to bridge the gap between nonlinear systems techniques and high-order, complex network models. The search for important network structures in this project is guided particularly by biochemical reaction networks which have evolved into several efficient circuit patterns in gene regulation, cell signaling, and biosynthetic pathways.
The first main goal in this project is to pursue a passivity-based approach to biochemical reaction networks. We have demonstrated this approach in  for cyclic networks, where the end product of a sequence of reactions inhibits the first reaction upstream, and obtained a nonlinear stability criterion that encompasses and significantly strengthens a local stability criterion used in mathematical biology for cyclic networks. In  we further generalized this passivity-based stability criterion and made it applicable to other network topologies, such as several branched pathways in metabolic networks. The next research topic is verification of passivity properties in network models using several numerical techniques, and modification of the proposed stability criteria to account for time delays. The final topic is the use of passivity properties to ensure robustness against diffusion-driven instabilities in spatially distributed models. Preliminary results in this direction have been reported in [3,4].
The second main goal of the project is to develop a novel model reduction technique that takes advantage of the structure of the network, rather than relying on its parameters. This technique is applicable to clustered structures such as the one in Figure 1, and exploits a two-time scale property whereby the clusters act as aggregate nodes that determine the dynamic behavior in the slow time-scale. Unlike earlier studies for power systems where the two time-scale property is due to the differing strength of the interconnection terms, we have revealed in  that the network structure alone can induce a time-scale separation even when the interconnection terms are of comparable strength. Clustered networks have been reported in cellular signaling networks, cell-cycle regulatory networks, and in metabolic networks. They further arise in sensor networks when communication is performed hierarchically through designated hubs in each cluster. The remaining research tasks include extending the analysis for undirected graphs in our preliminary work  to directed graphs which have wider practical applicability. This extension will be particularly useful for the order reduction of Markov chains, and will be applied to stochastic models of biochemical reaction networks which are preferable to deterministic models at low molecular concentrations.
Figure 1: A directed graph that exhibits clusters with sparse external links. The singular perturbation analysis in  shows that the dense connections cause coherent behavior to emerge within the clusters in the fast time scale. The clusters then behave as aggregate nodes that determine the dynamic behavior of the system in the slow time scale.
- Murat Arcak and Eduardo Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica, 42(9):1531-1537, 2006.
- Murat Arcak and Eduardo Sontag. A passivity-based stability criterion for a class of biochemical reaction networks. Mathematical Biosciences and Engineering, 5(1):1-19, 2008.
- Mihailo Jovanovic, Murat Arcak, and Eduardo Sontag. A passivity-based approach to stability of spatially distributed systems with a cyclic interconnection structure. IEEE Transactions on Automatic Control, 53(1):75-86, 2008.
- Xiaoqing Ge and Murat Arcak. A sufficient condition for additive D-stability and application to reaction-diffusion models. Systems and Control Letters, vol.58, no.2, pages 736-741, 2009.
- Emrah Biyik and Murat Arcak. Area aggregation and time scale modeling for sparse nonlinear networks. Systems and Control Letters, vol.57, no.2, pages 142-149, 2008.