  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.
