  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.
