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.

Diffusion is a fundamental graph process, underpinning such phenomena as epidemic disease contagion and the spread of innovation by word-of-mouth. We address the algorithmic problem of finding a set of k initial seed nodes in a network so that the expected size of the resulting cascade is maximized, under the standard independent cascade model of network diffusion. Runtime is a primary consideration for this problem due to the massive size of the relevant input networks. We provide a fast algorithm for the influence maximization problem, obtaining the near-optimal approximation factor of (1 - 1/e - epsilon), for any epsilon > 0, in time O((m+n)k log(n) / epsilon^2). Our algorithm is runtime-optimal (up to a logarithmic factor) and substantially improves upon the previously best-known algorithms which run in time Omega(mnk POLY(1/epsilon)). Furthermore, our algorithm can be modified to allow early termination: if it is terminated after O(beta(m+n)k log(n)) steps for some beta < 1 (which can depend on n), then it returns a solution with approximation factor O(beta). Finally, we show that this runtime is optimal (up to logarithmic factors) for any beta and fixed seed size k.

Over the last decade we have witnessed the rapid proliferation of large-scale complex networks, spanning many social, information and technological domains. While many of the tasks which users of such networks face are essentially global and involve the network as a whole, the size of these networks is huge and the information available to users is only local. In this dissertation we show that even when faced with stringent locality constraints, one can still effectively solve prominent algorithmic problems on such networks. In the first part of the dissertation we present a natural algorithmic framework designed to model the behaviour of an external agent trying to solve a network optimization problem with limited access to the network data. Our study focuses on local information algorithms --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. We address both network coverage problems as well as network search problems. Our results include local information algorithms for coverage problems whose performance closely match the best possible even when information about network structure is unrestricted. We also 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 network structure, but with any less information it is impossible to achieve a reasonable approximation. For preferential attachment networks, we obtain polylogarithmic approximations to the problem of finding the smallest subgraph that connects a subset of nodes and the problem of finding the highest-degree nodes. This is achieved by addressing a decade-old open question of BollobAis and Riordan on locally finding the root in a preferential attachment process. In the second part of the dissertation we focus on designing highly time efficient local algorithms for central mining problems on complex networks that have been in the focus of the research community over a decade: finding a small set of influential nodes in the network, and fast ranking of nodes. Among our results is an essentially runtime-optimal local algorithm for the influence maximization problem in the standard independent cascades model of information diffusion and an essentially runtime-optimal local algorithm for the problem of returning all nodes with PageRank bigger than a given threshold. Our work demonstrates that locality is powerful enough to allow efficient solutions to many central algorithmic problems on complex networks. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Computer and Information Science This dissertation is available at ScholarlyCommons: http://repository.upenn.edu/edissertations/737 First Advisor Michael Kearns Second Advisor Sanjeev Khanna

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.

Diffusion is a fundamental graph process, underpinning such phenomena as epidemic disease contagion and the spread of innovation by word-of-mouth. We address the algorithmic problem of finding a set of k initial seed nodes in a network so that the expected size of the resulting cascade is maximized, under the standard independent cascade model of network diffusion. Runtime is a primary consideration for this problem due to the massive size of the relevant input networks. We provide a fast algorithm for the influence maximization problem, obtaining the near-optimal approximation factor of (1 - 1/e - epsilon), for any epsilon > 0, in time O((m+n)k log(n) / epsilon^2). Our algorithm is runtime-optimal (up to a logarithmic factor) and substantially improves upon the previously best-known algorithms which run in time Omega(mnk POLY(1/epsilon)). Furthermore, our algorithm can be modified to allow early termination: if it is terminated after O(beta(m+n)k log(n)) steps for some beta < 1 (which can depend on n), then it returns a solution with approximation factor O(beta). Finally, we show that this runtime is optimal (up to logarithmic factors) for any beta and fixed seed size k.

A fundamental problem arising in many applications in Web science and social network analysis is, given an arbitrary approximation factor $c>1$, to output a set $S$ of nodes that with high probability contains all nodes of PageRank at least $\Delta$, and no node of PageRank smaller than $\Delta/c$. We call this problem {\sc SignificantPageRanks}. We develop a nearly optimal, local algorithm for the problem with runtime complexity $\tilde{O}(n/\Delta)$ on networks with $n$ nodes. We show that any algorithm for solving this problem must have runtime of ${\Omega}(n/\Delta)$, rendering our algorithm optimal up to logarithmic factors. Our algorithm comes with two main technical contributions. The first is a multi-scale sampling scheme for a basic matrix problem that could be of interest on its own. In the abstract matrix problem it is assumed that one can access an unknown {\em right-stochastic matrix} by querying its rows, where the cost of a query and the accuracy of the answers depend on a precision parameter $\epsilon$. At a cost propositional to $1/\epsilon$, the query will return a list of $O(1/\epsilon)$ entries and their indices that provide an $\epsilon$-precision approximation of the row. Our task is to find a set that contains all columns whose sum is at least $\Delta$, and omits any column whose sum is less than $\Delta/c$. Our multi-scale sampling scheme solves this problem with cost $\tilde{O}(n/\Delta)$, while traditional sampling algorithms would take time $\Theta((n/\Delta)^2)$. Our second main technical contribution is a new local algorithm for approximating personalized PageRank, which is more robust than the earlier ones developed in \cite{JehW03,AndersenCL06} and is highly efficient particularly for networks with large in-degrees or out-degrees. Together with our multiscale sampling scheme we are able to optimally solve the {\sc SignificantPageRanks} problem.

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 Bollobás 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 provider's decision of how much structure to make visible to its users can have a significant effect on a user's ability to interact strategically with the network.

Social and other networks have been shown empirically to exhibit high edge clustering — that is, the density of local neighborhoods, as measured by the clustering coefficient, is often much larger than the overall edge density of the network. In social networks, a desire for tight-knit circles of friendships — the colloquial “social clique” — is often cited as the primary driver of such structure. We introduce and analyze a new network formation game in which rational players must balance edge purchases with a desire to maximize their own clustering coefficient. Our results include the following: Several of our results hold even for weaker notions of equilibrium, such as those based on link stability.