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(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 θ(log 2 n) delivery time. Moreover, we show that the expected diameter of the graph is θ(log n), a log n facto

Modeling small-worlds with geographical factors: distance-bias & bounded-growth neighborhoods

Abstract. Studies of many real world networks show that geographical factors play a significant role. However, existing models for small-world properties and power-law degrees don’t fully consider these geographical factors. We propose a general model for small-world and power-law features which also considers geographical factors, including the distance-bias tendency (links tend to favor closer distances) and the property of bounded growth in neighborhood expansion. Our refined model combines a growth bounded base graph with a distance-bias distribution of random links. We show when the small-world effect may occur and how the diameter changes depending on the coordination between the distance-bias parameter and the two bounded growth parameters. This helps explain why the Internet graph is considered as a small-world with low diameter, but is locally growth bounded. We develop analysis techniques for graphs with non-uniform random links, including a fractal-based analysis. We also discuss future work on applications to network design, where our models help augment networks for improving routing and other related issues.

Certifying Data from Multiple Sources

Data integrity can be problematic when integrating and organizing information from many sources. In this paper we describe efficient mechanisms that enable a group of data owners to contribute data sets to an untrusted third-party publisher, who then answers users' queries. Each owner gets a proof from the publisher that his data is properly represented, and each user gets a proof that the answer given to them is correct. This allows owners to be confident that their data is being properly represented and for users to be confident they are getting correct answers. We show that a group of data owners can efficiently certify that an untrusted third party publisher has computed the correct digest of the owners' collected data sets. Users can then verify that the answers they get from the publisher are the same as a fully trusted publisher would provide, or detect if they are not. The results presented support selection and range queries on multi-attribute data sets and are an extension of earlier work on Authentic Publication which assumed that a single trusted owner certified all of the data