Nesting of dynamic systems and mode-dependent networks

For many networks, the connection pattern (often called the topology) can vary in time, depending on the changing state, or mode, of the modules within the network. For example, "airplane mode" is the name for one communicative mode of a modern cellphone, in which it will not connect with any cellphone towers; thus the topology of the cellular network is dependent on the modes of its modules. This paper addresses the issue of nesting such mode-dependent networks, in which a local network can be abstracted as a single module in a larger network. Each module in the network represents a dynamic system, whose behavior includes repeatedly updating its communicative mode. It is in this way that the dynamics of the modules controls the topology of the networks at all levels. This paper provides a formal semantics, using the category-theoretic framework of operads and their algebras, to capture the nesting property and dynamics of mode-dependent networks. We provide a detailed running example to ground the mathematics.