Complexity-size scaling relation in flow networks away from thermodynamic equilibrium: variational principles

Complex systems can be represented by flow networks for energy and matter such as Rayleigh-Benard convection, river basins, lightning, circulatory systems of different kinds, etc. Agents along those networks form flows avoiding obstacles and searching for path of least action, with a curvature described by the metric tensor. The action efficiency can be used as a numerical measure for its level of organization. The constraints for motion curve the space and they push the agents away from them, they can be modeled with a repulsive potential, and the nodes such as the source and sink can be modeled with an attractive potential. The flows do work on the constraints to motion reducing them and thus reducing the curvature. This process obeys the Gauss Principle of Least Constraint, the Hertz's Principle of Least Curvature, and ultimately the Principle of Least Action. The more organized the system, the shorter are the paths, the higher is its degree of organization i.e. complexity. The decrease in internal entropy of the system corresponds to increase of the external entropy production by faster transmission of matter and energy across its boundaries. In our Agent Based Modelling simulations of flow networks, the average path length measured, which depends on the size of the system, given by the number of agents. The curvature of the paths of agents between the source and the sink decreases with self-organization time, but, also with increasing number of agents.