Tight Bounds on the Ratio of Network Diameter to Average Internode Distance

The average internode distance for a network is fairly difficult to derive. There is often no closed-form formula for this parameter, leading to the need for simulation-based derivation methods. Network diameter, by contrast, is relatively easier to determine and, for many networks of common interest, we have closed-form formulas for it. The bounds established in this paper show that the two parameters are usually not totally independent and that, from a practical standpoint, network diameter can be used in lieu of average internode distance for the evaluation of message-routing algorithms and assessment or comparison of communication performance, particularly for symmetric networks that are prevalent in many parallel systems.