Graph-theoretic description of the interplay between non-linearity and connectivity in biological systems

The purpose of this article is to stress the implications that the consideration of nonlinearity has upon the extension and strength of connectivity, if this is understood as a characterization of the degree of interrelation between parts of the system. This objective is reached within the QP formalism for non-linear ODEs. The formalism is developed in a graph-theoretic setting, with the help of which the connectionist aspect of non-linearity becomes apparent. Topology-preserving transformations involve an exchange between the degree of non-linearity and the strengths of interactions, thus assembling systems of apparently different nature into classes of equivalence. We argue that, if we have in mind a classification of systems according to behavior, these classes of equivalence should be given their proper singularity. We characterize globally the connectivity of a class with an index, although we point out during the discussion that the mathematical conception of the complex idea of connectivity is still incomplete.