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×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(log2 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 θ(log2 n) delivery time. Moreover, we show that the expected diameter of the graph is θ(log n), a log n factor smaller. We also extend our results to the general kdimensional model. Our diameter results extend traditional w...

• Type: Conference
• Year: 2004
• Where: PODC
• Authors: Charles U. Martel, Van Nguyen
• Keywords: Distant Nodes, Distributed And Parallel Computing, Kleinberg’s Small-world Model, PODC 2004, Show