Physical Structure, Graph Structure and Uncertainty in Complex Systems
Mezic, UC Santa Barbara
Abstract: Complexity of systems can be roughly represented on 2 axes,
one parametrizing complexity of structure, the other complexity of
system dynamics. These two are often not independent, and disentangling
their interdependence is not trivial. I will begin by addressing
complexity of the system in the context of its behaviours under
parameter and initial condition uncertainty in the framework of ergodic
theory. Then I will discuss how graph-theoretic decompositions on state
variables allow for an extension of that analysis to higher dimensional
systems. In this approach, it is not the statistics of the graph that
plays the dominant role, but its functional submodules. Algorithms for
finding such submodules involve a combination of graph theory, control
theory and dynamical systems theory.
Biomolecular systems represent a rich source of interesting complex
systems behaviour but are not easily decomposed using above mentioned
graph theoretic methods on the standard position-velocity description.
I will discuss features of dynamics of a specific model on various
scales and the underlying spectral theory that allows for reduced
representations of a model biomolecular system that concurrently
exhibits robustness and flexibility. An interesting phenomenon occurs,
where specific structured perturbations effect global change is system
conformation, while non-structured ones do not affect its stability.
This enables the system to perform its function in an uncertain
environment. The phenomenon is driven by a resonant interaction and
captured accurately by a suitable spectral projection on a small number
of modes. In this system there is no separation of scales, but a
finite-dimensional representation is possible.
Finally I will discuss approaches to computation of uncertainty and
present a modulation of Quasi-Monte Carlo method using a construction
that samples the space of parameters or initial conditions using
uniformly ergodic dynamical systems that preserve the specified prior.