In this paper, we consider the problem of calculating fast and accurate approximations to the personalized PageRank score of a webpage. We focus on techniques to improve speed by limiting the amount of web graph data we need to access.

Our algorithms provide both the approximation to the personalized PageRank score as well as guidance in using only the necessary information—and therefore sensibly reduce not only the computational cost of the algorithm but also the memory and memory bandwidth requirements. We report experiments with these algorithms on web graphs of up to 118 million pages and prove a theoretical approximation bound for all. Finally, we propose a local, personalized web-search system for a future client system using our algorithms.

In this paper we suggest evaluating the importance of a website with the mean frequency of visiting the website for the Markov chain on the Internet graph describing random surfing. We show that this mean frequency is equal to the sum of the PageRanks of all the webpages in that website (hence referred to as PageRankSum), and we propose a novel algorithm, AggregateRank, based on the theory of stochastic complement to calculate the rank of a website. The AggregateRank algorithm gives a good approximation of the PageRankSum accurately, while the corresponding computational complexity is much lower than PageRankSum. By constructing return-time Markov chains restricted to each website, we describe also the probabilistic relation between PageRank and AggregateRank. The complexity of the AggregateRank algorithm, the error bound of the estimation, and the experiments are discussed at the end of the paper.

Motivated by copying models of the web graph, Bonato and Janssen introduced the following simple construction: given a graph $G$, for each vertex $x$ and each subset $X$ of its closed neighborhood, add a new vertex $y$ whose neighbors are exactly $X$. Iterating this construction yields a limit graph $\uparrow G$. Bonato and Janssen claimed that the limit graph is independent of $G$, and it is known as the infinite locally random graph. We show that this picture is incorrect: there are in fact infinitely many isomorphism classes of limit graph, and we give a classification. We also consider the inexhaustibility of these graphs.

In the valley-free path model, a path in a given directed graph is valid if it consists of a sequence of forward edges followed by a sequence of backward edges. This model is motivated by routing policies of autonomous systems in the Internet. We give a $2$-approximation algorithm for the problem of computing a maximumn number of edge- or vertex-disjoint valid paths between two given vertices $s$ and $t$, and we show that no better approximation ratio is possible unless $P=NP$. Furthermore, we give a $2$-approximation algorithm for the problem of computing a minimum vertex cut that separates $s$ and $t$ with respect to all valid paths and prove that the problem is APX-hard. The corresponding problem for edge cuts is shown to be polynomial-time solvable. For the multiway variant of the cut problem, we give a $4$-approximation algorithm. We present additional results for acyclic graphs

There is much current interest in the evolution of social networks, in particular, the World Wide Web graph, through time. ``Preferential attachment'' and the ``copying model'' are well-known models that explain the observed power-law degree distribution of the graph reasonably well. However, existing evolution models do not include the significant influence of search engines on how webpage authors find existing pages and create links to them. Recent applied work has raised the concern that highly popular search engines limit the attention of authors to a small set of ``celebrity'' URLs, for any query. Page authors frequently (with probability $p$) locate pages using a search engine. Then they link to popular pages among those they visit. We initiate an analysis of this more realistic process, show that the celebrity nodes eventually accumulate a constant fraction of all links created with high probability (whp), and show that the degrees of the other nodes still follow a power-law distribution, but with a steeper power: $\Pr(\mathrm{degree}=k)\propto k^{-(1+2/(1-p))}$ \whp. Our analysis adds evidence to the recent concern that search engines offer new webpages a steep, self-sustaining barrier to entry to well-connected, entrenched web communities.