AIMS Mathematics, 2017, 2(3): 437-450. doi: 10.3934/Math.2017.2.437

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Relations between the dynamics of network systems and their subnetworks

1 Department of Mathematics, Texas Southern University, 3100 Cleburne, Houston, TX, 77004, USA
2 Department of Mathematics, University of South Florida at St. Pete, 140 7th Avenue South St.Petersburg, Florida 33701, USA

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Statistical analysis of the connectivity of real world networks have revealed interesting featuressuch as community structure, network motif and as on. Such discoveries tempt us to understandthe dynamics of a complex network system by studying those of its subnetworks. This approach isfeasible only if the dynamics of the subnetwork systems can somehow be preserved or partially preservedin the whole system. Most works studied the connectivity structures of networks while very fewconsidered the possibility of translating the dynamics of a subnetwork system to the whole system. Inthis paper, we address this issue by focusing on considering the relations between cycles and fixedpoints of a network system and those of its subnetworks based on Boolean framework. We proved thatat a condition we called agreeable, if X0 is a fixed point of the whole system, then X0 restricted to thephase-space of one of the subnetwork systems must be a fixed point as well. An equivalent statementon cycles follows from this result. In addition, we discussed the relations between the product of thetransition diagrams (a representation of trajectories) of subnetwork systems and the transition diagramof the whole system.
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