Information geometry of scaling expansions of non-exponentially growing configuration spaces

Abstract

Many stochastic complex systems are characterized by the fact that their configuration space doesn't grow exponentially as a function of the degrees of freedom. The use of scaling expansions is a natural way to measure the asymptotic growth of the configuration space volume in terms of the scaling exponents of the system. These scaling exponents can, in turn, be used to define universality classes that uniquely determine the statistics of a system. Every system belongs to one of these classes. Here we derive the information geometry of scaling expansions of sample spaces. In particular, we present the deformed logarithms and the metric in a systematic and coherent way. We observe a phase transition for the curvature. The phase transition can be well measured by the characteristic length r, corresponding to a ball with radius 2r having the same curvature as the statistical manifold. Increasing characteristic length with respect to size of the system is associated with sub-exponential sample space growth is associated to strongly constrained and correlated complex systems. Decreasing of the characteristic lenght corresponds to super-exponential sample space growth that occurs for example in systems that develop structure as they evolve. Constant curvature means exponential sample space growth that is associated with multinomial statistics, and traditional Boltzmann-Gibbs, or Shannon statistics applies. This allows us to characterize transitions between statistical manifolds corresponding to different families of probability distributions.

A Bimodal Weibull Distribution: Properties and Inference

“…log q is q-deformed logarithm of natural logarithm (log) in MLE [19]. The concavity property of log q guarantees to replace log in MLE by log q [18,23,35,36].…”

An overview of generalized entropic forms

“…In this case, a statistical description based on entropic forms (11) or (18) fails to make a correct prediction. To overcame this lack, in [42,43] a generalization of (18) has been advanced…”

New probability distribution describing emergence in state space

“…Here we will concentrate on the statistics of a minimal model such of cases, namely the pairing model introduced in [2]. The paring model has been studied in the context of relating the N dependence of W (N ) to generalised entropies in a number of publications, see [3,4,5,6,7,8,9,10,11,12]. One may think of the components of the paring model as coins.…”