Can an overlay compensate for a careless underlay?

We investigate the ability of an overlay network to compensate for "careless" routing in the native network layer, i.e., for network-layer routes not optimized for performance. We consider cost/traffic-dependent, overlay routing on top of an underlay routing based on randomly-generated, cost/traffic-independent link weights, and determine the extent to which overlay-overcareless-underlay can achieve performance close to that attainable when underlay routing is performed in an optimal cost/traffic-dependent ("careful") manner. We identify three graph-theoretic metrics that collectively characterize the richness of the underlay network: the tightness (characteristic path length, or CPL), the thickness (cut average) and the weighted sum of node degrees. We find that only when the underlay graph is rich can the overlay compensate for a careless underlay. If network links have linear costs, we prove that overlay cost is a linear increasing function of CPL under some assumptions. Finally, we investigate heuristics (based on the notion of richness) for choosing a set of overlay nodes of fixed size that is likely to result in good overlay performance.