A local partitioning algorithm finds a set with small conductance near a specified seed vertex. In this paper, we present a generalization of a local partitioning algorithm for undirected graphs to strongly connected directed graphs. In particular, we prove that by computing a personalized PageRank vector in a directed graph, starting from a single seed vertex within a set $S$ that has conductance at most $\alpha$, and by performing a sweep over that vector, we can obtain a set of vertices $S'$ with conductance $\Phi_{M}(S')= O(\sqrt{\alpha \log |S|})$. Here, the conductance function $\Phi_{M}$ is defined in terms of the stationary distribution of a random walk in the directed graph. In addition, we describe how this algorithm may be applied to the PageRank Markov chain of an arbitrary directed graph, which provides a way to partition directed graphs that are not strongly connected.

Motivated by the problem of detecting link-spam, we consider the following graph-theoretic primitive: Given a webgraph $G$, a vertex $v$ in $G$, and a parameter $\delta \in (0,1)$, compute the set of all vertices that contribute to $v$ at least a $\delta$ fraction of $v$'s PageRank. We call this set the $\delta$-contributing set of $v$. To this end, we define the contribution vector of $v$ to be the vector whose entries measure the contributions of every vertex to the PageRank of $v$. A local algorithm is one that produces a solution by adaptively examining only a small portion of the input graph near a specified vertex. We give an efficient local algorithm that computes an $\epsilon$-approximation of the contribution vector for a given vertex by adaptively examining $O(1/\epsilon)$ vertices. Using this algorithm, we give a local approximation algorithm for the primitive defined above. Specifically, we give an algorithm that returns a set containing the $\delta$-contributing set of $v$ and at most $O(1/\delta)$ vertices from the $\delta/2$-contributing set of $v$, and which does so by examining at most $O(1/\delta)$ vertices. We also give a local algorithm for solving the following problem: If there exist $k$ vertices that contribute a $\rho$-fraction to the PageRank of $v$, find a set of $k$ vertices that contribute at least a $(\rho-\epsilon)$-fraction to the PageRank of $v$. In this case, we prove that our algorithm examines at most $O(k/\epsilon)$ vertices.

We study the PageRank mass of principal components in a bow-tie web graph as a function of the damping factor $c$. It is known that the web graph can be divided into three principal components: SCC, IN, and OUT. The giant strongly connected component (SCC) contains a large group of pages having a hyperlink path connecting them. The pages in the IN (OUT) component have a path to (from) the SCC, but not back. Using a singular perturbation approach, we show that the PageRank share of the IN and SCC components remains high even for very large values of the damping factor, in spite of the fact that it drops to zero when $c$ tends to one. However, a detailed study of the OUT component reveals the presence of ``dead ends'' (small groups of pages linking only to each other) that receive an unfairly high ranking when $c$ is close to $1$. We argue that this problem can be mitigated by choosing $c$ as small as $\frac12$.

Popular reputation systems for linked networks can be manipulated by spammers who strategically place links. In Sergey Brin and Lawrence Page, “The Anatomy of a Large-Scale Hypertextual Web Search Engine.” Computer Networks and ISDN Systems 30:1–7 (1998), 107–117, pages endorse others by placing links, and the global link structure is analyzed to determine the reputation of each page. Though this is meant to be a global measure, page $v$ can boost its own PageRank considerably using a simple self-endorsement strategy: placing outlinks to form short directed cycles. In contrast, we show that expected hitting time---the time to reach $v$ in a random walk---measures essentially the same quantity as PageRank, but does not depend on $v$'s outlinks. We develop a reputation system based on hitting time and show that it resists tampering by individuals or groups who strategically place outlinks. We also present an algorithm to efficiently compute hitting time for all nodes in a massive graph; conventional algorithms do not scale adequately.

In this paper we consider three models for random graphs that utilize the inner product as their fundamental object. We analyze the behavior of these models with respect to clustering, the small world property, and degree distribution. These models are motivated by the random dot product graphs developed by Kraetzl, Nickel, and Scheinerman. We extend their results to fully parameterize the conditions under which clustering occurs, characterize the diameter of graphs generated by these models, and describe the behavior of the degree distribution

We analyze the structure of random graphs generated by the geographical threshold model. The model is a generalization of random geometric graphs. Nodes are distributed in space, and edges are assigned according to a threshold function involving the distance between nodes as well as randomly chosen node weights. We show how the degree distribution, percolation and connectivity transitions, clustering coefficient, and diameter relate to the threshold value and weight distribution. We give bounds on the threshold value guaranteeing the presence or absence of a giant component, connectivity and disconnectivity of the graph, and small diameter. Finally, we consider the clustering coefficient for nodes with a given degree $l$, finding that its scaling is very close to $1/l$ when the node weights are exponentially distributed.

We study a general framework for decentralized search in random graphs. Our main focus is on deterministic memoryless search algorithms that use only local information to reach their destination in a bounded number of steps in expectation. This class includes (with small modifications) the search algorithms used in Kleinberg's pioneering work on long-range percolation graphs and hierarchical network models. We give a characterization of searchable graphs in this model, and use this characterization to prove a monotonicity property for searchability.

Social networks are ubiquitous. The discovery of closely knit clusters in these networks is of fundamental and practical interest. Existing clustering criteria are limited in that clusters typically do not overlap, all vertices are clustered, and/or external sparsity is ignored. We introduce a new criterion that overcomes these limitations by combining internal density with external sparsity in a natural way.

This paper explores combinatorial properties of internally dense and externally sparse clusters. A simple algorithm is given for provably finding such clusters assuming a sufficiently large gap between internal density and external sparsity. Experimental results show that the algorithm is able to identify over 90% of the clusters in real graphs, assuming conditions on external sparsity.

We present a new stochastic model for complex networks, based on a spatial embedding of the nodes, called the spatial preferred attachment (SPA) model. In the SPA model, nodes have influence regions of varying size, and new nodes may link to a node only if they fall within its influence region. The spatial embedding of the nodes models the background knowledge or identity of the node, which will influence its link environment. In our model, nodes can determine their link environment based only on local knowledge of the network. We prove that our model gives a power-law in-degree distribution, with exponent in $[2,\infty)$ depending on the parameters, and with concentration for a wide range of in-degree values.