In this paper, we introduce simple graph clustering methods based on minimum cuts within the graph. The clustering methods are general enough to apply to any kind of graph but are well suited for graphs where the link structure implies a notion of reference, similarity, or endorsement, such as web and citation graphs. We show that the quality of the produced clusters is bounded by strong minimum cut and expansion criteria. We also develop a framework for hierarchical clustering and present applications to real-world data. We conclude that the clustering algorithms satisfy strong theoretical criteria and perform well in practice.

We develop a coupling technique for analyzing online models by using offline models. This method is especially effective for a growth-deletion model that generalizes and includes the preferential attachment model for generating large complex networks which simulate numerous realistic networks. By coupling the online model with the offline model for random power law graphs, we derive strong bounds for a number of graph properties including diameter, average distances, connected components, and spectral bounds. For example, we prove that a power law graph generated by the growth-deletion model almost surely has diameter $ O(\log n)$ and average distance $ O(\log \log n)$.

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time $t$, with probability $\alpha_1>0$ we add a new vertex $u_t$ and $m$ random edges incident with $u_t$. The neighbours of $u_t$ are chosen with probability proportional to degree. With probability $\alpha-\alpha_1\geq 0$ we add $m$ random edges to existing vertices where the endpoints are chosen with probability proportional to degree. With probability $1-\alpha-\alpha_0$ we delete a random vertex, if there are vertices left to delete. With probability $\alpha_0$ we delete $m$ random edges. Assuming that $\alpha+\alpha_1+\alpha_0>1$ and $\alpha_0$ is sufficently small, we show that for large $k,t$, the expected number of vertices of degree $k$ is approximately $d_kt$ where as $k\to\infty$, $d_k\sim Ck^{-1-\beta}$ where $\beta=\frac{2(\alpha-\alpha_0)}{3\alpha-1-\alpha_1-\alpha_0}$ and $C>0$ is a constant. Note that $\beta$ can take any value greater than 1.

A Bloom filter is a simple space-efficient randomized data structure for representing a set in order to support membership queries. Bloom filters allow false positives but the space savings often outweigh this drawback when the probability of an error is controlled. Bloom filters have been used in database applications since the 1970s, but only in recent years have they become popular in the networking literature. The aim of this paper is to survey the ways in which Bloom filters have been used and modified in a variety of network problems, with the aim of providing a unified mathematical and practical framework for understanding them and stimulating their use in future applications.