The PageRank model pioneered by Google is the most common approach for generating web search results. We present a two-stage algorithm for computing the PageRank vector where the algorithm exploits the lumpability of the underlying Markov chain. We make three contributions. First, the algorithm speeds up the PageRank calculation significantly. With web graphs having millions of webpages, the speed-up is typically in the two- to three-fold range. The algorithm can also embed other acceleration methods such as quadratic extrapolation, the Gauss-Seidel method, or the Biconjugate gradient stable method for an even greater speed-up; cumulative speed-up is as high as 7 to 14 times. The second contribution relates to the handling of dangling nodes. Conventionally, dangling nodes are included only towards the end of the computation. While this approach works reasonably well, it can fail in extreme cases involving aggressive personalization. We prove that our algorithm is the generally correct way of handling dangling nodes using probabilistic arguments. We also discuss variants of our algorithm, including a multistage extension for calculating a generalized version of the PageRank model where different personalization vectors are used for webpages of different classes. The ability to form class associations may be useful for building more refined models of web traffic.

We study the two most common types of percolation processes on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability $p$, and afterwards we focus on site percolation where the vertices are retained with probability $p$. We establish critical values for $p$ above which a giant component emerges in both cases. Moreover, we show that, in fact, these coincide. As a special case, our results apply to power-law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.

We introduce and study a class of exchangeable random graph ensembles. They can be used as statistical null models for empirical networks, and as a tool for theoretical investigations. We provide general theorems that characterize the degree distribution of the ensemble graphs, together with some features that are important for applications, such as subgraph distributions and kernel of the adjacency matrix. A particular case of directed networks with power-law out–degree is studied in more detail, as an example of the flexibility of the model in applications.

Start with graph $G_0 \equiv \{V_1,V_2\}$ with one edge connecting the two vertices $V_1$, $V_2$. Now create a new vertex $V_3$ and attach it (i.e., add an edge) to $V_1$ or $V_2$ with equal probability. Set $G_1 \equiv \{V_1,V_2,V_3\}$. Let $G_n \equiv \{V_1,V_2,\ldots,V_{n+2}\}$ be the graph after $n$ steps, $n \geq 0$. For each $i$, $1 \leq i \leq n+2$, let $d_n(i)$ be the number of vertices in $G_n$ to which $V_i$ is connected. Now create a new vertex $V_{n+3}$ and attach it to $V_i$ in $G_n$ with probability proportional to $w(d_n(i))$, $1 \leq i \leq n+2$, where $w(\cdot)$ is a function from $N \equiv \{1,2,3,\ldots\}$ to $(0,\infty)$. In this paper, some results on behavior of the degree sequence $\{d_n(i)\}_{n\geq 1,i\geq 1}$ and the empirical distribution $\{\pi_n(j) \equiv \frac{1}{n} \sum^n_{i=1} I(d_n(i) = j)\}_{n\geq 1}$ are derived. Our results indicate that the much discussed power-law growth of $d_n(i)$ and power law decay of $\pi(j) \equiv \lim_{n \rightarrow \infty} \pi_n(j)$ hold essentially only when the weight function $w(\cdot)$ is asymptotically linear. For example, if $w(x) = cx^2$ then for $i\geq 1$, $\lim_n d_n(i)$ exists and is finite with probability (w.p.)\ 1 and $\pi(j) \equiv \delta_{j1}$, and if $w(x) = cx^p$, $1/2 <p < 1$ then for $i \geq 1$, $d_n(i)$ grows like $(\log n)^q$ where $q=(1-p)^{-1}$. The main tool used in this paper is an embedding in continuous time of pure birth Markov chains.

Pursuit-evasion games, such as the game of Cops and Robbers, are a simplified model for network security. In this game, cops try to capture a robber loose on the vertices of the network. The minimum number of cops required to win on a graph $G$ is its number. We present asymptotic results for the game of Cops and Robbers played in various stochastic network models, such as in $G(n,p)$ with nonconstant $p$ and in random power-law graphs. We find bounds for the cop number of $G(n,p)$ for a large range $p$ as a function of $n$. We prove that the cop number of random power-law graphs with $n$ vertices is asymptotically almost surely $\Theta(n)$. The cop number of the core of random power-law graphs is investigated, and it is proved to be of smaller order than the order of the core.