Asymptotic Information-Theoretic Detection of Dynamical Organization in Complex Systems

Abstract

The identification of emergent structures in complex dynamical systems is a formidable challenge. We propose a computationally efficient methodology to address such a challenge, based on modeling the state of the system as a set of random variables. Specifically, we present a sieving algorithm to navigate the huge space of all subsets of variables and compare them in terms of a simple index that can be computed without resorting to simulations. We obtain such a simple index by studying the asymptotic distribution of an information-theoretic measure of coordination among variables, when there is no coordination at all, which allows us to fairly compare subsets of variables having different cardinalities. We show that increasing the number of observations allows the identification of larger and larger subsets. As an example of relevant application, we make use of a paradigmatic case regarding the identification of groups in autocatalytic sets of reactions, a chemical situation related to the origin of life problem.

A Fast and Effective Method to Identify Relevant Sets of Variables in Complex Systems

“…The whole method (choice of an index based on entropic observations, its normalization, application of the iterated sieving algorithm-possibly by using at each stage only groups of relatively small size-until the final groups cannot be further expanded) is called here "full RI method" where RI stands for "Relevance Index" (in [21] we present several indices that can be used within the RI method, and we evaluate their effects and advantages). An example of application of the RI method is presented in Appendix D. For the sake of simplicity, in the following we will refer to the algorithm that implements this method, using the zI index, as the "full zI algorithm" (a version of the software, developed in Python language, is available at the link: https://github.com/gianlucadaddese/Iterative-zI, accessed on 23 April 2021).…”

“…The knowledge of the theoretical distribution of the integration values of such a system would, however, allow to avoid this load. In [21] it was shown that the quantity 2mI (m being the number of observations) follows a Chi Square distribution, whose freedom degrees are a function of the number of variables of the subset and on the cardinality of their alphabet.…”

“…A simple implementation of the iterative sieving algorithms corresponds therefore to subsequently merging a set of patterns into a new cluster for each iteration, according to the zI, and iterating the procedure, considering each time the merged variables as a new atomic entity. The final cluster set is composed by the groups, generated by such iterative mergers, that have been detected at the time of the last iteration, i.e., when the zI falls below a given threshold value zI_theta The recommended value of the threshold (zI_theta = 3.0) derives from statistical considerations and corresponds to a normalized distance of 3 standard deviations from the reference condition of variable independence [13,21]. These final groups are called Relevant Sets (RSs).…”

“…The lines show the fraction of times the given strategy has reported an ARI exactly equal to 1.0 or greater than 0.8 and the average of the ARI index. We considered the analysis of attractors in Boolean systems with topology and dynamics derived from real genetic regulatory networks (T_helper system, [39]), the propagation of transient perturbations in continuous dynamical systems (CSTR system, [14,15,21]), and permanent perturbations in genetic regulatory networks (knock-out experiments, this paper and [13,40,41]), the analysis of stochastic systems (Leader-Follower systems, [15]), and the analysis of social systems ("Green Community" case, described in [13]).…”

“…In order to avoid this pitfall, we also introduced an "iterative sieving" algorithm which leads to the identification of disjoint RSs, which can in turn be combined to obtain larger relevant sets. The method has been widely described in the literature [13,15,21] and is briefly summarized in Section 2 below. For the sake of clarity, it will be called the "full RI method" in this paper.…”