On the topological properties of grid-based interconnection networks: Surface area and volume of radial spheres

Grid-based networks (or grids for short), such as meshes and tori, have been the underlying topology for many multicomputers, and have been extensively studied in the past as a graph topology. In this paper, we investigate some topological properties of grids without boundary wrap-around (meshes) and with boundary wrap-around (tori). In particular, we study the problem of finding the number of nodes located at/within a given distance from a given node (surface area/volume) in the network and derive some expressions for computing such a number. Furthermore, we provide similar expressions that improve on previous results already reported in the literature for some special cases of grids, notably hypercubes and k-ary n-cubes. We also show some applications of the derived expressions in analytical performance modelling of some grid-based networks under uniform and hotspot traffic loads.