Routing in Strongly Hyperbolic Unit Disk Graphs
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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(log^2n + logk)) bits at each vertex and takes 𝒪(k) time for a routing decision, where k = π e (1 + a)/(2(b - 1)) (b^2 diam(G) - 1) R + log_b(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(log^2n) holds asymptotically almost surely. Thus, we obtain a worst-case stretch of 3, 𝒪(log^4 n) bits of storage per vertex, and 𝒪(log^2 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.
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