Complex Systems

by Georgi Yordanov Georgiev

Quantity of organization in complex networks here is measured as the inverse of the average sum of physical actions of all elements per unit motion multiplied by the Planck's constant. The meaning of quantity of organization is the inverse of the number of quanta of action per one unit motion of an element. This definition can be applied to the organization of any complex system. Systems self-organize to decrease the average action per element per unit motion. This lowest action state is the attractor for the continuous self-organization and evolution of a dynamical complex system. Constraints increase this average action and constraint minimization by the elements is a basic mechanism for action minimization. Increase of quantity of elements in a network, leads to faster constraint minimization through grouping, decrease of average action per element and motion and therefore accelerated rate of self-organization. Progressive development, as self-organization, is a process of minimization of action.

In complex systems, the interplay between nonlinear and stochastic dynamics gives rise to an evolution process in Darwinian sense with punctuated equilibrium, random "mutations" and "adaptations". The emergent discrete states in such a system, i.e., attractors, have natural robustness against both internal and external perturbations. Epigenetic states of a biological cell, a mesoscopic nonlinear stochastic open biochemical system, could be understood through such a framework.

by Nuno Crokidakis, Marcio Argollo de Menezes

In this work we study a modified Susceptible-Infected-Susceptible (SIS) model in which the infection rate $\lambda$ decays exponentially with the number of reinfections $n$, saturating after $n=l$. We find a critical decaying rate $\epsilon_{c}(l)$ above which a finite fraction of the population becomes permanently infected. From the mean-field solution and computer simulations on hypercubic lattices we find evidences that the upper critical dimension is 6 like in the SIR model, which can be mapped in ordinary percolation.