Efficient hierarchical analysis of the stability of a network through dimensional reduction of its influence topology

Abstract

The connection between network topology and stability remains unclear. General approaches that clarify this relationship and allow for more efficient stability analysis would be desirable. Inspired by chemical reaction networks, I demonstrate the utility of expressing the governing equations of arbitrary first-order dynamical systems (interaction networks) in terms of sums of real functions (generalized reactions) multiplied by real scalars (generalized stoichiometries). Specifically, I examine the mathematical notion of influence topology, which is based on the reaction stoichiometries and the first derivatives of the reactions with respect to each species at the steady state solution(s). It is naturally represented as a signed directed bipartite graph with arrows or blunt arrows connecting a species node to a reaction node (positive/negative derivative) or a reaction node to a species node (positive/negative stoichiometry). The set of all such graphs is denumerable. A significant reduction in dimensionality is possible through stoichiometric scaling, cycle compaction, and temporal scaling. All cycles in a network can be read directly from the graph of its influence topology, enabling efficient and intuitive computation of the principal minors (sums of products of non-overlapping bipartite cycles) and the Hurwitz determinants (sums of products of either the principal minors or the bipartite cycles) for testing steady state stability. The stability of a given network is shown to have a hierarchical dependence first on its influence topology and then, more specifically, on algebraic conditions (exact functional form of the reactions). The utility of this hierarchical approach to bifurcation analysis is demonstrated on classical networks from control theory, biology, chemistry, physics, and electronics.