Dynamics of Geographical Routing in Small-World Networks

A graph exhibits the small-world phenomenon if most nodes can be reached through a short chain of steps along the edges. Current small-world models include those based on a simple base graph structure, which is augmented with shortcuts added according to some random process. We examined the small-world graph models mentioned above with variable population density through computer simulations. Firstly, graphs were examined with a grid structure as base graph but allowing arbitrarily many nodes at each grid spot. Secondly, graphs were examined with continuous location coordinates. Finally, graphs based on real world population density data from Statistics Sweden were examined. Explicit distance based shortcut selection, presented by Kleinberg in 1999, works surprisingly well in all settings but the one based on real world population density data, where it fails to produce navigable graphs. Rank based friendship, presented by Liben-Nowell et al in 2005, is effective in producing navigable graphs in all settings examined, as is the destination sampling rewiring algorithm presented by Sandberg in 2005. However, the latter technique gives shorter routing times overall.