Directed and Almost-Directed Flow Loops in Real Networks

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. Keywords—directed flow loops; almost-directed flow loops; flow networks; optimization; classical algorithms; maximising the flow. I. DIRECTED LOOPS OF FLOW IN NETWORKS The existence of routing loops have already been reported in computer networks [1,2]. Due to inconsistencies in routing state among a set of routers, the packets physically travel along a closed loop and never reach their destination. Surprisingly, directed loops of commodity may exist even if none of the dispatched commodities physically travels along a closed loop. This point is illustrated by the examples in Fig.1 featuring supply networks (e.g. supply of petrol from a number of fuel terminals to a number of filling stations), where the same exchangeable commodity is transported along straight lines which are the shortest paths from sources to destinations. Selecting the shortest paths for a data transfer for example, is also a common strategy in communication networks [3]. Suppose that the throughput capacity of each sourcedestination straight-line path is 10 units. Despite that none of the dispatched commodities physically travels along a closed contour, a directed loop carrying 10 units of flow effectively appears between the intersection points (real or imaginary) x1,x2 and x3 in the network from Fig.1a and between nodes x1,x2,x3 and x4 in the network from Fig.1b. Removing 10 units of flow from the segments (x1,x2), (x2,x3) and (x3,x4) in Fig.1a and from the segments x1,x2), (x2,x3), (x3,x4) and (x4,x1) in Fig.1b turns the flow circulating along the contours x1,x2,x3,x1 and x1,x2,x3,x4,x1 into zero, without diminishing the amount of total flow sent from the source nodes to destination nodes (Fig.1c,1d). Figure 1 shows that directed loops of flow can even be found in networks where the intersecting source-destination paths are straight-line segments and no transported commodity physically travels along a closed contour. A closed contour formed by a sequence of n nonempty sections ( 3  n ), in which the flows point along the direction of traversing the contour will be referred to as “directed flow loop”.