Operads as a potential foundation for systems of systems

David Spivak (March 22, 2016)

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Rather than considering each node in a network as an atomic entity of some sort, it is often useful to regard the node as potentially consisting of a networked system in its own right. In this talk I will propose the mathematical theory of operads as a foundation for modeling the sort of nested structures that arise when considering the relationship between wholes and parts in a system of systems. The main case of interest will be open dynamical systems (e.g., ODEs or boolean networks), which can be interconnected together to form larger-scale dynamical systems. I will explain how operads can help us more easily calculate invariants of such systems, such as their steady state matrices (which generalize bifurcation diagrams). In particular, I will discuss how the steady state matrix of an interconnected system can be calculated using standard matrix operations (multiplication, tensor product, and trace) applied to the steady state matrices of its component systems.