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 non-linearity 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.