Scaling in complex systems: A link between the dynamics of networks and growing interfaces

Abstract

We consider growing interfaces as dynamical networks whose nodes are the discrete points of the interface and the edges the physical interactions among them. We map the points of the interface formed at each time into a graph by means of a visibility algorithm. As the corresponding interfaces grow, their visibility graphs change over time. We show that the visibility graphs are all scale free for each time. We use the variance of the node degrees as a measure of the dynamical properties of these graphs. This magnitude reveals an unexpected scaling behaviour of these graphs in both the number of nodes and time. This enables to define three robust exponents that characterize any type of dynamics with more detail than the classical scaling analysis applied directly to the physical interfaces. To check the feasibility of this approach we study and classify six different dynamical processes and estimate their critical exponents. We conclude that the dynamics of physical systems far from equilibrium can be determined by its corresponding visibility network. Indeed, this methodology is able to discern among dynamical processes that hitherto have been classified in the same universality class according to the scaling analysis of their interfaces.