The multidimensional torus: analysis of average hop distance and application as a multihop lightwave network

The torus network, whose two-dimensional version is often referred to as the Manhattan Street Network (MSN), has received significant attention in the literature. Analysis of the hop distance properties for this network when its links are bidirectional is straightforward. However, the analysis of this network with unidirectional links is more complex, and the literature only provides the mean hop distance for the unidirectional version when the nodal degree is two. The authors provide a closed-form, analytical formula for the average hop distance in a three-dimensional torus network with unidirectional links, and hypothesize an approximate result for higher dimensions. The importance of this work stems from the fact that the torus is a useful candidate for the construction of an optical network with a multihop virtual topology based on wavelength division multiplexing (WDM). So far, it has not been possible to conduct fair comparisons between the torus and other multihop topologies with nodal degree greater than two, but the present work will now enable such comparisons