Network Class Superposition Analyses

Networks are often used to understand a whole system by modeling the interactions among its pieces. Examples include biomolecules in a cell interacting to provide some primary function, or species in an environment forming a stable community. However, these interactions are often unknown; instead, the pieces' dynamic states are known, and network structure must be inferred. Because observed function may be explained by many different networks (e.g., [Image: see text] for the yeast cell cycle process [1]), considering dynamics beyond this primary function means picking a single network or suitable sample: measuring over all networks exhibiting the primary function is computationally infeasible. We circumvent that obstacle by calculating the network class ensemble. We represent the ensemble by a stochastic matrix [Image: see text], which is a transition-by-transition superposition of the system dynamics for each member of the class. We present concrete results for [Image: see text] derived from Boolean time series dynamics on networks obeying the Strong Inhibition rule, by applying [Image: see text] to several traditional questions about network dynamics. We show that the distribution of the number of point attractors can be accurately estimated with [Image: see text]. We show how to generate Derrida plots based on [Image: see text]. We show that [Image: see text]-based Shannon entropy outperforms other methods at selecting experiments to further narrow the network structure. We also outline an experimental test of predictions based on [Image: see text]. We motivate all of these results in terms of a popular molecular biology Boolean network model for the yeast cell cycle, but the methods and analyses we introduce are general. We conclude with open questions for [Image: see text], for example, application to other models, computational considerations when scaling up to larger systems, and other potential analyses.