The Search for Candidate Relevant Subsets of Variables in Complex Systems

Abstract

In this paper we describe a method to identify "relevant subsets" of variables, useful to understand the organization of a dynamical system. The variables belonging to a relevant subset should have a strong integration with the other variables of the same relevant subset, and a much weaker interaction with the other system variables. On this basis, extending previous works on neural networks, an information-theoretic measure is introduced, i.e. the Dynamical Cluster Index, in order to identify good candidate relevant subsets. The method does not require any previous knowledge of the relationships among the system variables, but relies on observations of their values in time. We show its usefulness in several application domains, including: (i) random boolean networks, where the whole network is made of different subnetworks with different topological relationships (independent or interacting subnetworks); (ii) leader-follower dynamics, subject to noise and fluctuations; (iii) catalytic reaction networks in a flow reactor; (iv) the MAPK signaling pathway in eukaryotes. The validity of the method has been tested in cases where the data are generated by a known dynamical model and the Dynamical Cluster Index method is applied in order to uncover significant aspects of its organization; however it is important to stress that it can also be applied to time series coming from field data without any reference to a model. Given that it is based on relative frequencies of sets of values, the method could be applied also to cases where the data are not ordered in time. Several indications to improve the scope and effectiveness of the Dynamical Cluster Index to analyze the organization of complex systems are finally given.

Dynamical Criticality: Overview and Open Questions

“…It has been shown that information-theoretic measures make it possible to detect dynamical structures in complex systems [90,26,89]. The method is based on a measure called the dynamical cluster index and can detect subsets of variables that are tightly integrated among themselves and loosely interacting with the rest of the systems.…”

An Iterative Information‐Theoretic Approach to the Detection of Structures in Complex Systems

“…These subsets are possible candidates as higher-level entities for describing the organization of a system; they will be called relevant subsets (RSs, omitting the specification that they are initially just candidates). A quantitative measure, well suited for identifying them, is defined as follows (the presentation below follows the one given in [5]).…”

“…In previous works [4,5], we have introduced the Relevance Index (RI), a measure based on information theory that seems suitable for exploring the organization of complex systems. The RI makes it possible to identify, as components of a system, relevant sets of variables that show an integrated behaviour and interact more weakly with the rest of the system.…”

Identifying Critical States through the Relevance Index

“…Since these subsets are possible candidates as higher-level entities, to be used to describe the system organization, they will be called relevant subsets (omitting the specification that they are candidates). A quantitative measure, well suited for identifying them, is defined as follows (the presentation below follows the one given in [9]). …”

“…However, entropies scale with system size, so this requires considerable ingenuity. Following the original work of Tononi, an "RI method" has been developed for this purpose, where the variable is first normalized with respect to a reference case, and a statistical index is computed that allows meaningful comparisons of sets of different sizes [6,9,10]. However, quite often, these sets overlap, so the actual organization of the system remains opaque; for example, a variable may belong to a set of three variables and also to one of its four-element supersets, both endowed with fairly high values of the statistical index.…”