I am most grateful to physicist Ijaz Durrani for referring me to a paper recently published by Liu, Slotine and Barabasi in the Proceedings of the US National Academy of Sciences, addressing the observability of nonlinear dynamical systems.
The authors believe to have proven that their graphical approach (GA) leads to the isolation of a minimum set of sensors (i.e., a subset of the system’s internal variables) necessary and sufficient to describe the dynamics of a complex system.
For linear dynamical systems, the minimum sensor set derived from the GA would only be necessary and not sufficient. But for nonlinear dynamical systems the GA sensor set is also sufficient. According to the authors, this stems from the fact that, unlike linear systems, nonlinear systems contain zero or almost zero symmetries in the state variables.
Any symmetries in the state variables that leave the inputs, outputs, and all their derivatives invariant make the system unobservable (i.e., you can’t look at outputs and say something positive about the system’s state): a dynamical system with internal symmetries can have an infinite number of temporal trajectories that cannot be distinguished from each other by monitoring outputs.
A complex system, on the other hand, is more essential, it has a personality (no symmetries): and this is why its behavior can be captured by a subset of the internal variables, i.e., by monitoring only some outputs.
The paper does not offer rigorous proof of the sufficiency of the GA-selected sensors. The authors have simply run a total of circa 1000 numerical simulations in several complex domains (such as Michaelis–Menten, Lotka-Volterra, and Hindmarsh–Rose) and found the GA-selected subset to be a sufficient descriptor.
The graphical approach reduces observability (a dynamical problem) to a property of the static map of an inference diagram: and such maps are available for an increasing number of complex problems, like the three mentioned above.
The graph is obtained as follows.
Like in the life-sciences example offered in the paper, consider a number of chemical substances
A, B, C, D, …
some of which are reacting with each others. Reactions, i.e., will be of the kind
A+B+C –> D+F+J
D <–> E
and so on. You may therefore write, using mass-action kinetics, balance equations representing all reactions: the equations will contain the substances’ concentrations as variables (xA, xB, xC, xD, …) and a number of rate constants k1, k2, …, as many as there are reactions.
From there, an inference diagram is built by drawing a directed link
xi –> xj
if xj appears in the right-hand side of xi ‘s balance equation.
Then, strongly connected components or SCCs are identified as the largest subgraphs such that there is a directed path from each node to every other node in the subgraph. Among these, “root” SCCs are those SCCs that have no incoming edges. At least one node is chosen from each root SCC, to ensure observability of the whole system.
These findings are likely to benefit various domains of public interest, such as medicine or economics and other social sciences.
There also are several lessons here for pop-complexity fans to learn: e.g., complexity can be managed, and it can be done using a scientific instead of a fideistic or animistic approach.
Paolo Magrassi 2013 Creative Commons Attribution-Non-Commercial-Share Alike