From Logistic Networks to Social Networks

3Exact Methods Applied to the Flow Analysis of Topological Networks

3.1. Introduction

The flow modeling, described in Chapter 2, makes it possible to isolate two particular types of networks that lend themselves favorably to the use of previously resolved models, namely: additive flow networks and synchronized flow networks.

Additive flow networks respect the principle of the conservation of flows (Kirchhoff’s law) which pass through the nodes of the network. This excludes AND-type routing activities (specific to information flows), as well as consolidation/deconsolidation activities. In terms of Petri nets, an additive flow network is a state graph, characterized by the absence of several converging edges towards the same transition and by the absence of edges diverging from the same transition (Table 3.1). In this case, the convergence/divergence of multiple edges is carried out exclusively at places and expresses the principle of OR-type routing: the tokens arriving via multiple edges cross a common place, which are then inserted into one of the outgoing flows. As the tokens are neither assembled nor split as they pass through the common place, the flows are simply intertwined and Kirchhoff’s law applies. This is the case in a road network where the flows of vehicles converging at a crossroads intersect with each other before recomposing into output flows in accordance with the destinations. The same is true for certain production workshops, when the flow of products passing ...