Complete Limits of Flow Network based on Critical Flow: Concept, Model, Algorithm, Visualization, and Applications.

To reveal the complete limits of a flow network, this paper proposes the concept of critical flow. Firstly, it is found that a flow network is in a critical state when it reaches its maximum flow, and the network cannot afford any flow increase. In fact, not only maximum flow but many other network flows, i.e., blocking flows, indicate the critical states of a flow network. From a new viewpoint, the concept of critical flow is proposed, which unifies the maximum flow and the blocking flow. Secondly, the mathematical model, solution algorithms, and visualization method for critical flow are proposed. An analytical algorithm and a numerical algorithm are proposed to calculate the complete critical flows. In addition, a critical flow curve is designed to visualize the complete limits of a network, with the top point being the maximum flow. Finally, case studies are conducted to illustrate the critical flow and its typical applications in flow networks and real networks. It is concluded that the complete limits of a flow network are the range of critical flows, not a single maximum flow value. The proposed critical flows describe the complete limits of a flow network for the first time. Critical flow can be used for network monitoring and network planning. In monitoring, it reflects the margin of the flow network more accurately than maximum flow. In planning, it evaluates the flow transmission capability more accurately than maximum flow. [ABSTRACT FROM AUTHOR]