No 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 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 factor smaller. We also extend our results to the general kdimensional 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 n) 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 θ(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.

Query answers from on-line databases can easily be corrupted by hackers or malicious database publishers. Thus it is important to provide mechanisms which allow clients to trust the results from on-line queries. Authentic publication allows untrusted publishers to answer securely queries from clients on behalf of trusted off-line data owners. Publishers validate answers using hard-to-forge verification objects (VOs), which clients can check efficiently. This approach provides greater scalability, by making it easy to add more publishers, and better security, since on-line publishers do not need to be trusted.To make authentic publication attractive, it is important for the VOs to be small, efficient to compute, and efficient to verify. This has lead researchers to develop independently several different schemes for efficient VO computation based on specific data structures. Our goal is to develop a unifying framework for these disparate results, leading to a generalized security result. In this paper we characterize a broad class of data structures which we call Search DAGs, and we develop a generalized algorithm for the construction of VOs for Search DAGs. We prove that the VOs thus constructed are secure, and that they are efficient to compute and verify. We demonstrate how this approach easily captures existing work on simple structures such as binary trees, multi-dimensional range trees, tries, and skip lists. Once these are shown to be Search DAGs, the requisite security and efficiency results immediately follow from our general theorems. Going further, we also use Search DAGs to produce and prove the security of authenticated versions of two complex data models for efficient multi-dimensional range searches. This allows efficient VOs to be computed (size O(log N + T )) for typical one-and two-dimensional range queries, where the query answer is of size T and the database is of size N . We also show I/O-efficient schemes to construct the VOs. For a system with disk blocks of size B, we answer one-dimensional and three-sided range queries and compute the VOs with O(log B N + T /B) I/O operations using linear size data structures.

Integrity critical databases, such as financial information used in high-value decisions, are frequently published over the Internet. Publishers of such data must satisfy the integrity, authenticity, and non-repudiation requirements of clients. Providing this protection over public data networks is an expensive proposition. This is, in part, due to the difficulty of building and running secure systems. In practice, large systems can not be verified to be secure and are frequently penetrated. The negative consequences of a system intrusion at the publisher can be severe. The problem is further complicated by data and server replication to satisfy availability and scalability requirements.To our knowledge this work is the first of its kind to give general approaches for reducing the trust required of publishers of large databases. To do this, we separate the roles of data owner and data publisher. With a few digital signatures on the part of the owner and no trust required of a publisher, we give techniques based on Merkle hash trees that publishers can use to provide authenticity and non-repudiation of the answer to database queries posed by a client. This is done without requiring a key to be held in an on-line system, thus reducing the impact of system penetrations. By reducing the trust required of the publisher, our solution is a step towards the publication of large databases in a scalable manner.