Quantifying the complexity of flow networks: How many roles are there?

Weighted flow networks are structures that arise naturally when analyzing complex systems. The countable properties of unweighted networks are not easily generalized to weighted networks. One candidate measure of complexity is the number of roles, or specialized functions in a network. It is easy to identify the number of roles in a linear or cyclic unweighted network. There is only one logically consistent way to generalize the measures of nodes, flows, connectivity, and roles into weighted networks, and these generalizations are equivalent to indices derived from information theory and used by ecologists since the late seventies. Data from ecosystem networks suggests that ecosystems inhabit a narrow window of the parameter space defined by these measures.