Directed and Almost-Directed Flow Loops in Real Networks

Abstract

Directed flow loops are highly undesirable because they are associated with wastage of energy for maintaining them and entail big losses to the world economy. It is shown that directed flow loops may appear in networks even if the dispatched commodity does not physically travel along a closed contour. Consequently, a theorem giving the necessary and sufficient condition of a directed flow loop on randomly oriented straight-line flow paths has been formulated and a close-form expression has been derived for the probability of a directed flow loop. The results show that even for a relatively small number of intersecting flow paths, the probability of a directed flow loop is very large, which means that the existence of directed flow loops in real networks is practically inevitable. Consequently, a theorem and an efficient algorithm have been proposed related to discovering and removing directed flow loops in a network with feasible flows. The new concept ‘almost-directed flow loop’ has also been introduced for the first time. It is shown that the removal of an almost-directed flow loop also results in a significant decrease of the losses. It is also shown that if no directed flow loops exist in the network, the removal of an almost-directed flow loop cannot create a directed flow loop.