Michael Brautbar

The power of local information in social networks

We study the power of local information algorithms for optimization problems on social and technological networks. We focus on sequential algorithms where the network topology is initially unknown and is revealed only within a local neighborhood of vertices that have been irrevocably added to the output set. This framework models the behavior of an external agent that does not have direct access to the network data, such as a user interacting with an online social network. We study a range of problems under this model of algorithms with local information. When the underlying graph is a preferential attachment network, we show that one can find the root (i.e. initial node) in a polylogarithmic number of steps, using a local algorithm that repeatedly queries the visible node of maximum degree. This addresses an open question of Bollobas and Riordan. This result is motivated by its implications: we obtain polylogarithmic approximations to problems such as finding the smallest subgraph that connects a subset of nodes, finding the highest-degree nodes, and finding a subgraph that maximizes vertex coverage per subgraph size. Motivated by problems faced by recruiters in online networks, we also consider network coverage problems on arbitrary graphs. We demonstrate a sharp threshold on the level of visibility required: at a certain visibility level it is possible to design algorithms that nearly match the best approximation possible even with full access to the graph structure, but with any less information it is impossible to achieve a non-trivial approximation. We conclude that a network providers decision of how much structure to make visible to its users can have a significant effect on a users ability to interact strategically with the network.

A sublinear time algorithm for pagerank computations

In a network, identifying all vertices whose PageRank is more than a given threshold value Δ is a basic problem that has arisen in Web and social network analyses. In this paper, we develop a nearly optimal, sublinear time, randomized algorithm for a close variant of this problem. When given a directed network G=(V,E), a threshold value Δ, and a positive constant c>3, with probability 1−o(1), our algorithm will return a subset S⊆V with the property that S contains all vertices of PageRank at least Δ and no vertex with PageRank less than Δ/c. The running time of our algorithm is always

Multiscale Matrix Sampling and Sublinear-Time PageRank Computation

\tilde{O}(\frac{n}{\Delta})

On the Power of Adversarial Infections in Networks

. In addition, our algorithm can be efficiently implemented in various network access models including the Jump and Crawl query model recently studied by [6], making it suitable for dealing with large social and information networks. As part of our analysis, we show that any algorithm for solving this problem must have expected time complexity of

A clustering coefficient network formation game

{\Omega}(\frac{n}{\Delta})

Local Algorithms for Finding Interesting Individuals in Large Networks

. Thus, our algorithm is optimal up to logarithmic factors. Our algorithm (for identifying vertices with significant PageRank) applies a multi-scale sampling scheme that uses a fast personalized PageRank estimator as its main subroutine. For that, we develop a new local randomized algorithm for approximating personalized PageRank which is more robust than the earlier ones developed by Jeh and Widom [9] and by Andersen, Chung, and Lang [2].

Influence Maximization in Social Networks: Towards an Optimal Algorithmic Solution

Abstract A fundamental problem arising in many applications in Web science and social network analysis is the problem of identifying all nodes in a network whose PageRank exceeds a given threshold Δ. In this paper, we study the probabilistic version of the problem whereby given an arbitrary approximation factor c > 1, we are asked to output a set S of nodes such that with high probability, S contains all nodes of PageRank at least Δ, and no node of PageRank smaller than Δ/c. We call this problem SignificantPageRanks. We develop a nearly optimal local algorithm for the problem with time complexity on networks with n nodes, where the tilde hides a polylogarithmic factor. We show that every algorithm for solving this problem must have running time of Ω(n/Δ), rendering our algorithm optimal up to logarithmic factors. Our algorithm has sublinear time complexity for applications including Web crawling and Web search that require efficient identification of nodes whose PageRanks are above a threshold Δ = nδ, for some constant 0 < δ < 1. Our algorithm comes with two main technical contributions. The first is a multiscale sampling scheme for a basic matrix problem that could be of interest on its own. For us, it appears as an abstraction of a subproblem we need to tackle in order to solve the SignificantPageRanks problem, but we hope that this abstraction will be useful in designing fast algorithms for identifying nodes that are significant beyond PageRank measurements. In the abstract matrix problem, it is assumed that one can access an unknown right-stochastic matrix by querying its rows, where the cost of a query and the accuracy of the answers depend on a precision parameter ε. At a cost propositional to 1/ε, the query will return a list of O(1/ε) entries and their indices that provide an ε-precision approximation of the row. Our task is to find a set that contains all columns whose sum is at least Δ and omits every column whose sum is less than Δ/c. Our multiscale sampling scheme solves this problem with cost , while traditional sampling algorithms would take time Θ((n/Δ)2). Our second main technical contribution is a new local algorithm for approximating personalized PageRank, which is more robust than the earlier ones developed in [Jeh and Widom 03, Andersen et al. 06] and is highly efficient, particularly for networks with large in-degrees or out-degrees. Together with our multiscale sampling scheme, we are able to solve the SignificantPageRanks problem optimally.

Sublinear Time Algorithm for PageRank Computations and Related Applications

Over the last decade we have witnessed the rapid proliferation of online networks and Internet activity. Such activity is considered as a blessing but it brings with it a large increase in risk of computer malware — malignant software that actively spreads from one computer to another. To date, the majority of existing models of malware spread use stochastic behavior, when the set of neighbors infected from the current set of infected nodes is chosen obliviously. In this work, we initiate the study of adversarial infection strategies which can decide intelligently which neighbors of infected nodes to infect next in order to maximize their spread, while maintaining a similar “signature” as the oblivious stochastic infection strategy as not to be discovered. We first establish that computing an optimal and near-optimal adversarial strategies is computationally hard. We then identify necessary and sufficient conditions in terms of network structure and edge infection probabilities such that the adversarial process can infect polynomially more nodes than the stochastic process while maintaining a similar “signature” as the oblivious stochastic infection strategy. Among our results is a surprising connection between an additional structural quantity of interest in a network, the network toughness, and adversarial infections. Based on the network toughness, we characterize networks where existence of adversarial strategies that are pandemic (infect all nodes) is guaranteed, as well as efficiently computable.