Subnetwork inclusion and switching of multilevel Boolean networks preserve parameter graph structure and dynamics

This study addresses a problem of correspondence between dynamics of a parameterized system and the structure of interactions within that system. The structure of interactions is captured by a signed network. A network dynamics is parameterized by collections of multi-level monotone Boolean functions (MBFs), which are organized in a parameter graph PG . Each collection generates dynamics which are captured in a structure of recurrent sets called a Morse graph. We study two operations on signed graphs, switching and subnetwork inclusion, and show that these induce dynamics-preserving maps between parameter graphs. We show that duality, a standard operation on MBFs, and switching are dynamically related: If M is the switch of N , then duality gives an isomorphism between PG ( N ) and PG ( M ) which preserves dynamics and thus Morse graphs. We then show that for each subnetwork M ⊂ N , there are embeddings of the parameter graph PG ( M ) into PG ( N ) that preserve the Morse graph. Since our combinatorial description of network dynamics is closely related to switching ODE network models, our results suggest similar results for parameterized sets of smooth ODE network models of the network dynamics.