The preferential attachment graph is a random graph formed by adding a new vertex at each time-step, with a single edge which points to a vertex selected at random with probability proportional to its degree. Every $m$ steps the most recently added $m$ vertices are contracted into a single vertex, so at time $t$ there are roughly $t/m$ vertices and exactly $t$ edges. This process yields a graph which has been proposed as a simple model of the World Wide Web. For any constant $k$, let $\Delta_1 \geq \Delta_2 \geq \cdots \geq \Delta_k$ be the degrees of the $k$ highest degree vertices. We show that at time $t$, for any function $f$ with $f(t)\rightarrow\infty$ as $t\rightarrow\infty$, $\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq t^{1/2}f(t),$ and for $i = 2,\dots, k$, $\frac{t^{1/2}}{f(t)} \leq \Delta_i \leq \Delta_{i-1} - \frac{t^{1/2}}{f(t)},$ with high probability \whp). We use this to show that at time $t$ the largest $k$ eigenvalues of the adjacency matrix of this graph have $\lambda_k = (1\pm o(1))\Delta_k^{1/2}$ \whp.

We consider the following game introduced by Chung, Graham, and Leighton. One player, $\mathbf A$, picks $k \gt 1$ secrets from a universe of $N$ possible secrets, and another player, $\mathbf B$, tries to gain as much information about this set as possible by asking binary questions $f : [N] \rightarrow \{0,1\}$. Upon receiving a question $f$, $\mathbf A$ adversarially chooses one of her $k$ secrets, and answers $f$ according to it.

In this paper we present an explicit set of $2^{O(k)}(\log N)$ questions, along with a \[ $2^{O(k^2)}(\log^2 N)$ \] recovery algorithm that achieves $\mathbf B$'s goal in this game. This, in particular, completely solves the problem for any constant number of secrets $k$.

Our strategy is based on the list decoding of Reed-Solomon codes, and it extends and generalizes ideas introduced by Alon, Guruswami, Kaufman, and Sudan.

We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a self-adjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a max-degree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.

This survey reviews the research related to PageRank computing. Components of a PageRank vector serve as authority weights for web pages independent of their textual content, solely based on the hyperlink structure of the web. PageRank is typically used as a web search ranking component. This defines the importance of the model and the data structures that underly PageRank processing. Computing even a single PageRank is a difficult computational task. Computing many PageRanks is a much more complex challenge. Recently, significant effort has been invested in building sets of personalized PageRank vectors. PageRank is also used in many diverse applications other than ranking.

We are interested in the theoretical foundations of the PageRank formulation, in the acceleration of PageRank computing, in the effects of particular aspects of web graph structure on the optimal organization of computations, and in PageRank stability. We also review alternative models that lead to authority indices similar to PageRank and the role of such indices in applications other than web search. We also discuss link-based search personalization and outline some aspects of PageRank infrastructure from associated measures of convergence to link preprocessing.