analyzing kleinberg's (and other) small-world models
We analyze the properties of Small-World networks, where links are much more likely to connect "neighbor nodes" than distant nodes. In particular, our analysis provides new results for Kleinberg's Small-World model and its extensions. Kleinberg adds a number of directed long-range random links to an n x n lattice network (vertices as nodes of a grid, undirected edges between any two adjacent nodes). Links have a non-uniform distribution that favors arcs to close nodes over more distant ones. He shows that the following phenomenon occurs: between any two nodes a path with expected length O (log 2 n ) can be found using a simple greedy algorithm which has no global knowledge of long-range links.We show that Kleinberg's analysis is tight: his algorithm achieves ?(log 2 n ) delivery time. Moreover, we show that the expected diameter of the graph is ?log n ), a log n factor smaller. We also extend our results to the general k -dimensional model. Our diameter results extend traditional work on the diameter of random graphs which largely focuses on uniformly distributed arcs. Using a little additional knowledge of the graph, we show that we can find shorter paths: with expected length O (log 3/2 n ) in the basic 2-dimensional model and O (log 1+1/ k ) in the general k -dimensional model (for k ?1).Finally, we suggest a general approach to analyzing a broader class of random graphs with non-uniform edge probabilities. Thus we show expected ?(log n ) diameter results for higher dimensional grids, as well as settings with less uniform base structures: where links can be missing, where the probability can vary at different nodes, or where grid-related factors (e.g. the use of lattice distance) has a weaker role or is dismissed, and constraints (such as the uniformness of degree distribution) are relaxed.
model small-world network network links connection analysis extensibility direct randomization lattice grid distributed log use greedy algorithm algorithm global knowledge timing diameter graph factor general random graphs class probability distance role constraint routing small world