Routing in Strongly Hyperbolic Unit Disk Graphs

Greedy routing has been studied successfully on Euclidean unit disk graphs, which we interpret as a special case of hyperbolic unit disk graphs. While sparse Euclidean unit disk graphs exhibit grid-like structure, we introduce strongly hyperbolic unit disk graphs as the natural counterpart containing graphs that have hierarchical network structures. We develop and analyze a routing scheme that utilizes these hierarchies. On arbitrary graphs this scheme guarantees a worst case stretch of max{3, 1 + 2b/a} for a > 0 and b > 1. Moreover, it stores O(k (log2 n + logk)) bits at each vertex and takes O(k) time for a routing decision, where k = πe (1 + a)/(2(b − 1)) (b2 diam(G) − 1)R + logb (diam(G)) + 1, on strongly hyperbolic unit disk graphs with threshold radius R > 0. In particular, for hyperbolic random graphs, which have previously been used to model hierarchical networks like the internet, k = O(log2 n) holds asymptotically almost surely. Thus, we obtain a worst-case stretch of 3, O(log4 n) bits of storage per vertex, and O(log2 n) time per routing decision on such networks. This beats existing worst-case lower bounds. Our proof of concept implementation indicates that the obtained results translate well to real-world networks.