Efficient Routing in Networks with Long Range Contacts

Navigating Low-Dimensional and Hierarchical Population Networks

“…More generally, our results can be viewed as extending Kleinberg's theorem to a dimension-independent model that allows varying population density (and one that holds in real networks [21]). There have been some recent theoretical results extending and refining Kleinberg's result-for example, considering routing on other types of underlying graphs [28,7,9], among other results [4,23,26]-and we might hope to be able to make analogous improvements to our results.…”

“…There have been several relevant extensions to Kleinberg's original model, which we review here. In k-dimensional grids, there has also been considerable work on upper and lower bounds for the diameter and the length of the greedy path (e.g., [23,26,4]), and partially decentralized algorithms other than greedy routing have also been considered [8,20,22,23,27]. Kleinberg has extended his model to tree-based structures and group structures [17].…”

Eclecticism shrinks even small worlds

“…(The same holds for [13]). Greedy [7] O( 1 c log 2 n) O(c log n) Greedy [3,13] ( 1 c log 2 n) O(c log n) Greedy [2] ( 1 c log 2 n/ log log n) O(c log n) NoN-greedy [12] O( 1 c log c log 2 n) O(c 2 log n) Decentralized algorithm [10] …”

“…Surprisingly however, Kleinberg's model does not reflect this fact, in the sense that greedy routing has the same performance whether the number of mesh dimensions considered is one, two, or more. Indeed, Kleinberg has shown that greedy routing in the n-node d-dimensional mesh augmented with long-range links chosen according to the d-harmonic distribution performs in O(log 2 n) expected number of steps, i.e., independently of d. (This bound is tight as it was shown in [3] that greedy routing performs in at least (log 2 n) expected number of steps, independently of d). Kleinberg has also shown that augmenting the d dimensional mesh with the r -harmonic distribution, r = d, results in poor performance, i.e., (n α r ) expected number of steps for some positive constant α r .…”

“…Obviously, if nodes are given more than one long-range contact, then the performance of greedy routing can be improved, however to a limited extent only. For instance, with c long-range contacts per node, Kleinberg's greedy routing would perform in ( 1 c log 2 n) expected number of steps [3], which remains (log 2 n) for c = O (1). We propose a model in which the log 2 n barrier can be overcome, with a constant number c (say, c = 1) of long-range contacts per node.…”