We study the link structure of online social networks (OSNs) and introduce a new model for such networks that may help in inferring their hidden underlying reality. In the geo-protean (GEO-P) model for OSNs, nodes are identified with points in Euclidean space, and edges are stochastically generated by a mixture of the relative distance of nodes and a ranking function. With high probability, the GEO-P model generates graphs satisfying many observed properties of OSNs, such as power-law degree distributions, the small-world property, densification power law, and bad spectral expansion. We introduce the dimension of an OSN based on our model and examine this new parameter using actual OSN data. We discuss how the geo-protean model may eventually be used as a tool to group users with similar attributes using only the link structure of the network.

Network-creation games have been studied recently in many different settings. These games are motivated by social networks in which selfish agents want to construct a connection graph among themselves. Each node wants to minimize its average or maximum distance to the others, without paying much to construct the network. Many generalizations have been considered, including nonuniform interests between nodes, general graphs of allowable edges, and bounded-budget agents. In all of these settings, there is no known constant bound on the price of anarchy. In fact, in many cases, the price of anarchy can be very large, namely, a constant power of the number of agents. This means that we have no control over the behavior of a network when agents act selfishly. On the other hand, the price of stability in all these models is constant, which means that there is chance that agents act selfishly and we end up with a reasonable social cost.

In this paper, we show how to use an advertising campaign (as introduced in [Balcan et al., “Improved Equilibria via Public Service Advertising.”) to find efficient equilibria in an $(n, k)$-uniform bounded-budget connection game; our result holds for $k = \omega(\log(n))$. More formally, we present advertising strategies such that if an $\alpha$ fraction of the agents agree to cooperate in the campaign, the social cost will be at most $O(1/\alpha)$ times the optimum cost. This is the first constant bound on the price of anarchy that interestingly can be adapted to different settings. We also generalize our method to work in cases in which $\alpha$ is not known in advance. Also, we do not need to assume that the cooperating agents spend all their budget on the campaign; even a small fraction ($\beta$ fraction) of their budget is sufficient to obtain a constant price of anarchy.

We give algorithms for finding graph clusters and drawing graphs, highlighting local community structure within the context of a larger network. For a given graph G, we use the personalized PageRank vectors to determine a set of clusters, by optimizing the jumping parameter α subject to several cluster variance measures in order to capture the graph structure according to PageRank. We then give a graph visualization algorithm for the clusters using PageRank-based coordinates. Several drawings of real-world data are given, illustrating the partition and local community structure.

We explore methods for approximating the commute time and Katz score between a pair of nodes. These methods are based on the approach of matrices, moments, and quadrature developed in the numerical linear algebra community. They rely on the Lanczos process and provide upper and lower bounds on an estimate of the pairwise scores. We also explore methods to approximate the commute times and Katz scores from a node to all other nodes in the graph. Here, our approach for the commute times is based on a variation of the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adapt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real-world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pairwise commute-time method and columnwise Katz algorithm both have attractive theoretical properties and empirical performance.

Networks are a powerful way to describe and represent social, technological, and biological systems, where nodes represent entities (people, web sites, genes) and edges represent interactions (friendships, communication, regulation). The study of such networks then seeks to find common structural patterns and explain their emergence through tractable models of network formation.

In most networks, each node is associated with a rich set of attributes or features. For example, users in online social networks have profile information, genes have properties and functions, and web pages contain text. However, most existing network models focus on modeling the network structure while ignoring the features and properties of the nodes. Thus, the questions that we address in this work are as follows: What is a mathematically tractable model that naturally captures ways in which the network structure and node attributes interact? What are the properties of networks that arise under such a model?

We present a model of network structure that we refer to as the multiplicative attribute graphs (MAG) model. The MAG model naturally captures the interactions between the network structure and the node attributes. We consider a model in which each node has a vector of categorical attributes associated with it. The link-affinity matrix then models the interaction between the value of a particular attribute and the probability of a link between a pair of nodes. The MAG model yields itself to mathematical analysis, and we derive thresholds for the connectivity and the emergence of the giant connected component, and show that the model gives rise to networks with a constant diameter. We also analyze the degree distribution and find surprising flexibility of the MAG model in that it can generate networks with either log-normal or power-law degree distribution.

The number of triangles is a computationally expensive graph statistic frequently used in complex network analysis (e.g., transitivity ratio), in various random graph models (e.g., exponential random graph model), and in important real-world applications such as spam detection, uncovering the hidden thematic structures in the Web, and link recommendation. Counting triangles in graphs with millions and billions of edges requires algorithms that run fast, use little space, provide accurate estimates of the number of triangles, and preferably are parallelizable. In this paper we present an efficient triangle-counting approximation algorithm that can be adapted to the semistreaming model. Its key idea is to combine the sampling algorithm of Tsourakakis, and the partitioning of the set of vertices into high- and low-degree subsets as in Alon, treating each set appropriately. From a mathematical perspective, we present a simplified proof of Tsourakakis, “Counting Triangles Using Projections,” that uses the powerful Kim–Vu concentration inequality based on the Hajnal–Szemerédi theorem. Furthermore, we improve bounds of existing triple-sampling techniques based on a theorem of Ahlswede and Katona. We obtain a running time $O(m + \frac{m^{3 / 2} \log n} {t\epsilon^2})$ and an $(1 ±\epsilon)$ approximation, where $n$ is the number of vertices, $m$ is the number of edges, and $\Delta$ is the maximum number of triangles in which any single edge is contained. Furthermore, we show how this algorithm can be adapted to the semistreaming model with space usage $O(m^{1/2} \log n + \frac{m^{3 / 2} \log n} {t\epsilon^2} )$ and a constant number of passes (three) over the graph stream. We apply our methods to various networks with several millions of edges and we obtain excellent results, outperforming existing triangle-counting methods. Finally, we propose a random-projection-based method for triangle counting and provide a sufficient condition to obtain an estimate with low variance.