Relations between the dynamics of network systems and their subnetworks

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Statistical analysis of the connectivity of real world networks have revealed interesting features such as community structure, network motif and as on. Such discoveries tempt us to understand the dynamics of a complex network system by studying those of its subnetworks. This approach is feasible only if the dynamics of the subnetwork systems can somehow be preserved or partially preserved in the whole system. Most works studied the connectivity structures of networks while very few considered the possibility of translating the dynamics of a subnetwork system to the whole system. In this paper, we address this issue by focusing on considering the relations between cycles and fixed points of a network system and those of its subnetworks based on Boolean framework. We proved that at a condition we called agreeable, if X0 is a fixed point of the whole system, then X0 restricted to the phase-space of one of the subnetwork systems must be a fixed point as well. An equivalent statement on cycles follows from this result. In addition, we discussed the relations between the product of the transition diagrams (a representation of trajectories) of subnetwork systems and the transition diagram of the whole system.