Many natural and man-made systems of interest to scientists and engineers are composed of groups of individual elements interacting with each other, through specific channels, to produce and regulate appropriate responses. Examples include chemical reaction networks, cellular (signaling, transcriptional, and metabolic) networks, pharmacokinetic networks, epidemiological networks, ecological networks, social networks, neural networks, multi-agent networks, etc. Understanding the fundamental properties and design principles of such networks is an exciting and challenging research problem whose solution requires development of new theoretical and computational approaches. To complicate matters, the dynamics of most interaction networks are inherently nonlinear and stochastic. A unifying approach is thus needed to concurrently encompass the deterministic and stochastic aspects of networks that can lead to common approaches and methods for the modeling and analysis of a diverse body of interaction networks.
The main goal of this research is to develop a general theoretical and computational approach for characterizing and analyzing complexity in nonlinear interaction networks with Markovian dynamics. While complexity may be difficult to characterize directly, this research rigorously pursues and quantifies the prevalent hallmarks of complexity: self-organization, functional stability, robustness, and evolutionary behavior. The investigators study a potential energy landscape perspective that relates topographic features of the landscape to these fundamental network properties. To achieve feasibility and computational efficiency of the main goals of this effort, new tools are being developed for computing the time evolution of the probability distribution of the underlying stochastic population dynamics. Rigorous mathematical, algorithmic, and computational approaches are then proposed for modeling the emergent behavior and complexity of Markovian interaction networks as well as for studying their functional stability and thermodynamic robustness by probabilistic sensitivity analysis techniques.
