Analyzing Kleinberg's (and other) small-world Models

Abstract

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 <i>n</i>x<i>n</i> 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 <i>O</i>(log<sup>2</sup><i>n</i>) 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 &#952;(log<sup>2</sup><i>n</i>) delivery time. Moreover, we show that the expected diameter of the graph is &#952;log <i>n</i>), a log <i>n</i> factor smaller. We also extend our results to the general <i>k</i>-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 <i>O</i>(log<sup>3/2</sup><i>n</i>) in the basic 2-dimensional model and <i>O</i>(log<sup>1+1/<i>k</i></sup>) in the general <i>k</i>-dimensional model (for<i>k</i>&#8805;1).Finally, we suggest a general approach to analyzing a broader class of random graphs with non-uniform edge probabilities. Thus we show expected &#952;(log <i>n</i>) 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.

DOI: 10.1145/1011767.1011794

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@inproceedings{Martel2004AnalyzingK, title={Analyzing Kleinberg's (and other) small-world Models}, author={Charles U. Martel and Van Nguyen}, booktitle={PODC}, year={2004} }