Polylogarithmic network navigability using compact metrics with small stretch

Graph augmentation theory is a general framework for analyzing navigability in social networks. It is known that, for large classes of graphs, there exist augmentations of these graphs such that greedy routing according to the shortest path metric performs in polylogarithmic expected number of steps. However, it is also known that there are classes of graphs for which no augmentations can enable greedy routing according to the shortest path metric to perform better than Ω(<i>n</i>^{1/√log <i>n</i>}) expected number of steps. In fact, the best known universal bound on the greedy diameter of arbitrary graph is essentially <i>n</i>^1/3. That is, for any graph, there is an augmentation such that greedy routing according to the shortest path metric performs in Õ(<i>n</i>^1/3) expected number of steps. Hence, greedy routing according to the shortest path metric has at least two drawbacks. First, it is in general space-consuming to encode locally the shortest path distances to all the other nodes, and, second, greedy routing according to the shortest path metric performs poorly in some graphs. We prove that, using semimetrics of small stretch results in a huge positive impact, in both encoding space and efficiency of greedy routing. More precisely, we show that, for any connected <i>n</i>-node graph <i>G</i> and any integer <i>k</i> ≥ 1, there exist an augmentation φ of <i>G</i> and a semimetric μ on <i>G</i> with stretch 2<i>/k</i>-1 such that greedy routing according to μ performs in <i>O</i>(<i>k</i>² <i>n</i>^2/<i>k</i>log²<i>n</i>) expected number of steps. As a corollary, we get that for any connected <i>n</i>-node graph <i>G</i>, there exist an augmentation φ of <i>G</i> and a semimetric μ on <i>G</i> with stretch <i>O</i>(log <i>n</i>) such that greedy routing according to μ performs in polylogarithmic expected number of steps. This latter semimetric can be encoded locally at every node using only a polylogarithmic number of bits.