Analysis of a network’s emerging behaviour via its structure involving its strongly connected components

In this chapter, it is addressed how network structure can be related to network behaviour. If such a relation is studied, that usually concerns only strongly connected networks and only linear functions describing the aggregation of multiple impacts. In this chapter both conditions are generalised. General theorems are presented that relate emerging behaviour of a network to the network’s structure characteristics. The network structure characteristics on the one hand concern network connectivity in terms of the network’s strongly connected components and their mutual connections; this generalises the condition of being strongly connected (as addressed in Chap. 11) to a very general condition. On the other hand, the network structure characteristics considered concern aggregation by generalising from linear combination functions to any combination functions that are normalised, monotonic and scalar-free, so that many nonlinear functions are also covered (which also was done in Chap. 11). Thus the contributed theorems generalise existing theorems on the relation between network structure and network behaviour that only address specific cases (such as acyclic networks, fully and strongly connected networks, and theorems addressing only linear functions).