While several analytic models aim to explain the existence of short paths in social networks such as the web, relatively few address the problem of efficiently finding them, especially in a decentralized manner. Since developing purely decentralized search algorithms in general social-network models appears hard, we relax the notion of decentralized search by allowing the option of storing a small amount of preprocessed information about the network. We show that one can identify a small set of vertices in an undirected social network so that connectivity information of the vertices in this set can be used in conjunction with the local connectivity properties to perform decentralized search and find short paths between vertices. Our results are for random graphs with power law degree distribution generated by a variant of the expected degree model.

We show that in power law random graphs, almost surely, the expected rate at which a random walk with lookahead discovers the nodes of the graph is sublinear.

Today's search engines are increasingly required to broaden their capabilities beyond free-text search. More complex features, such as supporting range constraints over numeric data, are becoming common; structured search over XML data will soon follow. This is particularly true in the enterprise search domain, where engines attempt to integrate data from the web and corporate knowledge portals with data residing in proprietary databases. In this paper we extend previous schemes by which an inverted-index-based search engine can efficiently support queries that contain numeric restrictions in addition to standard, free-text portions. Furthermore, we analyze both the known schemes and our extensions in terms of index-build time, index space, and query processing time. We show how to maximize query processing performance while respecting limits on index size and build time, or conversely, how to minimize index space and build time while maintaining guarantees on runtime performance. Thus, we concisely analyze the trade-off between index size and build time, and runtime performance. Finally, we present experimental results that demonstrate significant performance benefits attained by our method, as compared to alternative approaches.

We study a random graph $G_n$ that combines certain aspects of geometric random graphs and preferential attachment graphs. The vertices of $G_n$ are $n$ sequentially generated points $x_1,x_2,\ldots,x_n$ chosen uniformly at random from the unit sphere in $\mathbb{R}^3$. After generating $x_t$, we randomly connect that point to $m$ points from those points in $x_1,x_2,\ldots,x_{t-1}$ that are within distance $r$ of $x_t$. Neighbors are chosen with probability proportional to their current degree, and a parameter $\alpha$ biases the choice towards self loops. We show that if $m$ is sufficiently large, if $r\geq \ln n / n^{1/2-\beta}$ for some constant $\beta$, and if $\alpha>2$, then with high probabilty (\whp) at time $n$ the number of vertices of degree $k$ follows a power law with exponent $\alpha+1$. Unlike the preferential attachment graph, this geometric preferential attachment graph has small separators, similar to experimental observations of Blandford, et al. We further show that if $m\geq K\ln n$, for $K$ sufficiently large, then $G_n$ is connected and has diameter $O(\ln n/r)$ \whp.

PageRank is one of the principle criteria according to which Google ranks webpages. PageRank can be interpreted as a frequency of webpage visits by a random surfer, and thus it reflects the popularity of a webpage. In the present work we find an analytical expression for the expected PageRank value in a scale-free growing network model as a function of the age of the growing network and the age of a particular node. Then, we derive asymptotics that show that PageRank follows closely a power law in the middle range of its values. The exponent of the theoretical power law matches very well the value found from measurements of the World Wide Web. Finally, we provide a mathematical insight for the choice of the damping factor in PageRank definition.

We consider the problem of determining the unknown characteristics of a random routing strategy from end-to-end measurements. More specifically, we construct a Markov chain that models the traffic of messages in a multiterminal network consisting of input, intermediate, and output terminals. The topology of the network is assumed to be known, but the Markovian routing strategy is not known. We solve the problem of determining the unknown one-step transition probability matrix of our random walk from input-output measurements of ``travel time.'' We give explicit inversion formulas (up to a natural gauge) in a nontrivial example. The result holds for a large (but not arbitrary) class of multiterminal networks, many of which are indicated here. The networks that we display here are constructed in a canonical fashion from certain graphs. Some of these graphs as well as the way to go from graphs to networks are also displayed. One example of a graph for which our method works is the edge graph of a hypercube in any dimension.