Optimal transport on wireless networks

We present a study of the application of a variant of a recently introduced heuristic algorithm for the optimization of transport routes on complex networks to the problem of finding the optimal routes of communication between nodes on wireless networks. Our algorithm iteratively balances network traffic by minimizing the maximum node betweenness on the network. The variant we consider specifically accounts for the broadcast restrictions imposed by wireless communication by using a different betweenness measure. We compare the performance of our algorithm to two other known algorithms and find that our algorithm achieves the highest transport capacity both for minimum node degree geometric networks, which are directed geometric networks that model wireless communication networks, and for configuration model networks that are uncorrelated scale-free networks. The study of transport on complex networks has attracted a great deal of interest in recent years [1–22]. One of the most important problems is to determine the routes between pairs of nodes that optimize the efficiency of the transport. Oftentimes, the routes that are used on networks are the so-called shortest-path routes, which are the routes with the minimum number of hops between any two nodes. As the volume of the transport increases, this approach leads to congestion or jamming of highly connected nodes on the network called hubs. Interest has developed in finding the routes that allow a given network to bear the highest possible amount of traffic without jamming [2–7]. This is done in general by defining the length of a path as a sum of weights assigned to the links that form it, and then reweighting various links to spread the traffic throughout the network and avoid jamming at hubs. The problem of finding the exact optimal routing has been shown [3, 23] to be NP -hard. Recently, however, we introduced a new heuristic routing optimization algorithm and showed [5,6] that it finds near-optimal routing on both random (ErdősRényi) [24] and scale-free [25] networks. Remarkably, this algorithm runs in only polynomial time O(N logN). We also found that optimal routing preserves the small world character of networks [26]. In this paper we use a variant of our algorithm to find optimal routes for transport on wireless communication networks, which has been the subject of a number of recent papers [7–13]. Wireless networks are described by variants of random geometric networks. These networks lack the small world effect [26] that characterizes all of other types of networks we have previously applied our algorithm to. Transport on wireless networks occurs only along the subset of links that are bidirectional. However, in order to avoid broadcasting interference, every time a node broadcasts an information packet the nodes at the ends of all of its outgoing links, whether they are bidirectional or not, are prevented from broadcasting or receiving packets. The variant of our algorithm we consider here accounts for these broadcasting restrictions. In order to study the effectiveness of our optimization of routing on wireless networks and to understand how the optimization process works, we will compare results given by our optimal routing algorithm (OR) with those obtained using the shortest path routing algorithm (SP) and with the algorithm introduced in Ref. [7] (KR). The three algorithms will be applied to two different types of networks: minimum node degree geometric networks that, as stated above, are good models for wireless communication networks [13], and configuration model networks that are uncorrelated scale-free networks [27]. Note that recent studies have shown that scale-free distributions of the node degrees are achievable [28–30] on geometric networks