Compact yet Accurate Causal Mathematical Models of Complex Dynamics with Applications in Cyber-Physical Systems

From microbial communities, human physiology to social and biological/ neural networks, from information technology and connected vehicle networks, to smart power grid, complex interdependent systems display multi-scale spatio-temporal patterns that are classified as non-linear, non-Gaussian, non-ergodic, and/or fractal. Distinguishing between the sources of nonlinearity, identifying the nature of fractality (space versus time) and encapsulating their mathematical characteristics into dynamic compact causal models remains a major challenge for studying and optimizing complex systems.

To address these challenges, we propose a new mathematical modeling paradigm for constructing compact yet accurate models of complex systems dynamics that learns the causal effects and influences among cyber and physical processes by analyzing the statistics of the magnitude increments and the inter-event times of stochastic processes. Unlike current trends in nonlinear system modeling which postulate mathematical expressions (possibly without physical based feedback), this mathematical analysis of the magnitude increments and the inter-event times allows us to extract knowledge about the causal dynamics and the degree of nonlinearity among cyber and physical processes and encode the observed statistical properties through compact mathematical expressions (fewest parameters) and a multi-fractional order nonlinear partial differential equations for the probability to find the system in a specific state at one time. Extensive simulation experiments on several sets of physiological processes demonstrate that the derived mathematical models offer superior accuracy over state of the art techniques to model brain-muscle interdependency. Current efforts investigate the capabilities of this mathematical framework to model homeostasis and robustness and explain the principles of robust decision-making dynamics in biological systems.