Emergence in Network Systems is Bounded by Boundary-Crossing Paths

Abstract

Emergence, features of a whole that no part shows alone, is central to complex systems but rarely measured so as to locate its source. Viewing a system structurally, as a network of interacting parts, we evaluate emergence as what observing the coupled whole reveals beyond, or erases from, the combined observations of the parts. For observations that build what they see route by route, we show that every emergent feature is produced by its own route across the interface between parts, a route that traces it to the components and interactions responsible. We show the number of all such boundary-crossing paths bounds the total emergence discrepancy, and equals the emergence of measures that count routes. The bound is computed locally around the interface, validated through network examples, and in a nervous system the routes show that interneurons relay most of the emergence. Our work associates emergence with path structure in the network system that can be anticipated, attributed, and engineered.