NextBestOnce: Achieving Polylog Routing despite Non-greedy Embeddings

Social Overlays suffer from high message delivery delays due to insufficient routing strategies. Limiting connections to device pairs that are owned by individuals with a mutual trust relationship in real life, they form topologies restricted to a subgraph of the social network of their users. While centralized, highly successful social networking services entail a complete privacy loss of their users, Social Overlays at higher performance represent an ideal private and censorship-resistant communication substrate for the same purpose. Routing in such restricted topologies is facilitated by embedding the social graph into a metric space. Decentralized routing algorithms have up to date mainly been analyzed under the assumption of a perfect lattice structure. However, currently deployed embedding algorithms for privacy-preserving Social Overlays cannot achieve a sufficiently accurate embedding and hence conventional routing algorithms fail. Developing Social Overlays with acceptable performance hence requires better models and enhanced algorithms, which guarantee convergence in the presence of local optima with regard to the distance to the target. We suggest to model Social Overlays as graphs embedded in Zn with a scale-free degree distribution with exponent α. The inaccuracy of the embedding is measured by a parameter C. We then show that our previously introduced routing algorithm NextBestOnce achieves an expected routing length ofO ( logα−1 n log log n+ C3 log n ) on our Social Overlay model. A lower bound on the performance of NextBestOnce is given by Ω ( logα−1 n+ C ) . Furthermore, we show that leveraging information from the two-hop neighborhood, a Neighbor-of-Neighbor (NoN) modification of our algorithm achieves an expected routing length of O ( logδ(α)(α−1) n log logn+ C3 log n ) , where δ(α) < 1. Hence, NoN information can indeed be used to improve the asymptotic routing complexity by more than a constant factor.