The web may be viewed as a graph each of whose vertices corresponds to a static HTML web page and each of whose edges corresponds to a hyperlink from one web page to another. Recently, there has been considerable interest in using random graphs to model complex real-world networks to gain an insight into their properties. In this paper we propose an extended version of a new random model of the web graph in which the degree of a vertex depends on its age. We use the differential equation method to obtain basic results on the probability of edges being present. From this we are able to characterize the degree sequence of the model and study its behaviour near the connectivity threshold.

We consider the combination of two ideas from the hashing literature: the power of two choices and Bloom filters. Specifically, we show via simulations that, in comparison with a standard Bloom filter, using the power of two choices can yield modest reductions in the false positive probability using the same amount of space and more hashing. While the improvements are sufficiently small that they may not be useful in most practical situations, the combination of ideas is instructive; in particular, it suggests that there may be ways to obtain improved results for Bloom filters while using the same basic approach they employ, as opposed to designing new, more complex data structures for the problem.

A local graph partitioning algorithm finds a cut near a specified starting vertex, with a running time that depends largely on the size of the small side of the cut, rather than the size of the input graph. In this paper, we present a local partitioning algorithm using a variation of PageRank with a specified starting distribution. We derive a mixing result for PageRank vectors similar to that for random walks, and we show that the ordering of the vertices produced by a PageRank vector reveals a cut with small conductance. In particular, we show that for any set $C$ with conductance $\Phi$ and volume $k$, a PageRank vector with a certain starting distribution can be used to produce a set with conductance $O(\sqrt{\Phi \log k})$. We present an improved algorithm for computing approximate PageRank vectors, which allows us to find such a set in time proportional to its size. In particular, we can find a cut with conductance at most $\phi$, whose small side has volume at least $2^{b}$, in time $O(2^{b} \log^2m/\phi^2)$ where $m$ is the number of edges in the graph. By combining small sets found by this local partitioning algorithm, we obtain a cut with conductance $\phi$ and approximately optimal balance in time $O(m \log^4 m/\phi^2)$.

The problem of radio channel assignments with multiple levels of interference depending on distance can be modelled using graph theory. The authors previously introduced a model of labeling by real numbers. Given a graph $G$, possibly infinite, and real numbers $k_1,k_2\ge0$, an $L(k_1,k_2)$-labeling of $G$ assigns real numbers $f(x)\ge0$ to the vertices $x$, such that the labels of vertices $u$ and $v$ differ by at least $k_i$ if $u$ and $v$ are at distance $i$ apart. We denote by $\lambda(G;k_1,k_2)$ the infimum span over such labelings~$f$. It is enough to determine $\lambda(G;k,1)$ for reals $k\ge0$, which will be a continuous nondecreasing piecewise linear function. Here we present these functions for paths, cycles, and wheels.

We study a random graph $G_n$ that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power law degree distribution where the expansion property depends on a tunable parameter of the model.

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 R$^3$. After generating $x_t$, we randomly connect it to $m$ points from those points $x_1,x_2,\ldots,x_{t-1}$.

In this paper, we study the configuration model (CM) with independent and identically-distributed (i.i.d.) degrees. We establish a phase transition for the diameter when the power-law exponent $\tau$ of the degrees satisfies $\tau \in (2,3)$. Indeed, we show that for $\tau>2$ and when vertices with degree $1$ or $2$ are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for $\tau \in (2,3)$, the diameter of the graph is, with high probability, bounded from above by a constant times the $\log \log$ of the size of the graph.