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The Power of Local Information in Social Networks
, 2012
"... We study the power of local information algorithms for optimization problems on social and technological networks. We focus on sequential algorithms for which the network topology is initially unknown and is revealed only within a local neighborhood of vertices that have been irrevocably added to th ..."
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We study the power of local information algorithms for optimization problems on social and technological networks. We focus on sequential algorithms for which 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. The distinguishing feature of this setting is that locality is necessitated by constraints on the network information visible to the algorithm, rather than being desirable for reasons of efficiency or parallelizability. In this sense, changes to the level of network visibility can have a significant impact on algorithm design. This framework captures situations in which the optimizer is an external agent that does not have direct access to the network data, but rather learns about the graph structure only via (costly) queries. For instance, a user may wish to strategically find, and form connections to, highdegree nodes in an online social network. An appropriatealgorithm for this search problem must take into account the fact that the structure of the graph is not known in advance, and is only revealed locally as nodes are added to the user’s set of connections. Given this limited network visibility, how should the user choose which connections to form? This question is
Fast Iterative Graph Computation with Block Updates
"... Scaling iterative graph processing applications to large graphs is an important problem. Performance is critical, as data scientists need to execute graph programs many times with varying parameters. The need for a highlevel, highperformance programming model has inspired much research on graph pr ..."
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Scaling iterative graph processing applications to large graphs is an important problem. Performance is critical, as data scientists need to execute graph programs many times with varying parameters. The need for a highlevel, highperformance programming model has inspired much research on graph programming frameworks. In this paper, we show that the important class of computationally light graph applications – applications that perform little computation per vertex – has severe scalability problems across multiple cores as these applications hit an early “memory wall ” that limits their speedup. We propose a novel blockoriented computation model, in which computation is iterated locally over blocks of highly connected nodes, significantly improving the amount of computation per cache miss. Following this model, we describe the design and implementation of a blockaware graph processing runtime that keeps the familiar vertexcentric programming paradigm while reaping the benefits of blockoriented execution. Our experiments show that blockoriented execution significantly improves the performance of our framework for several graph applications. 1.
Effective resistances, statistical leverage, and applications to linear equation solving
 CoRR
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A local algorithm for finding wellconnected clusters
 CoRR
, 2013
"... Motivated by applications of largescale graph clustering, we study randomwalkbased local algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. In particular, we develop a method with better theoretical guarantee compared to all previous work, b ..."
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Cited by 7 (2 self)
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Motivated by applications of largescale graph clustering, we study randomwalkbased local algorithms whose running times depend only on the size of the output cluster, rather than the entire graph. In particular, we develop a method with better theoretical guarantee compared to all previous work, both in terms of the clustering accuracy and the conductance of the output set. We also prove that our analysis is tight, and perform empirical evaluation to support our theory on both synthetic and real data. More specifically, our method outperforms prior work when the cluster is wellconnected. In fact, the better it is wellconnected inside, the more significant improvement we can obtain. Our results shed light on why in practice some randomwalkbased algorithms perform better than its previous theory, and help guide future research about local clustering. 1.
Finding Small Sparse Cuts by Random Walk
"... Abstract. We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V,E) and a parameter k ≤ E, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph ..."
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Abstract. We study the problem of finding a small sparse cut in an undirected graph. Given an undirected graph G = (V,E) and a parameter k ≤ E, the small sparsest cut problem is to find a set S ⊆ V with minimum conductance among all sets with volume at most k. Using ideas developed in local graph partitioning algorithms, we obtain the following bicriteria approximation algorithms for the small sparsest cut problem: – If there is a set U ⊆ V with conductance φ and vol(U) ≤ k, then there isapolynomial timealgorithm tofindaset S with conductance O ( √ φ/ǫ) and vol(S) ≤ k 1+ǫ for any ǫ> 1/k. – If there is a set U ⊆ V with conductance φ and vol(U) ≤ k, then there isapolynomial timealgorithm tofindaset S with conductance O ( √ φlogk/ǫ) and vol(S) ≤ (1+ǫ)k for any ǫ> 2logk/k. These algorithms can be implemented locally using truncated random walk, with running time almost linear to k. 1
Towards an SDPbased Approach to Spectral Methods A NearlyLinearTime Algorithm for Graph Partitioning and Decomposition
"... In this paper, we consider the following graph partitioning problem: The input is an undirected graph G = (V, E), a balance parameter b ∈ (0, 1/2] and a target conductance value γ ∈ (0, 1). The output is a cut which, if nonempty, is of conductance at most O ( f), for some function f (G, γ), and whi ..."
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In this paper, we consider the following graph partitioning problem: The input is an undirected graph G = (V, E), a balance parameter b ∈ (0, 1/2] and a target conductance value γ ∈ (0, 1). The output is a cut which, if nonempty, is of conductance at most O ( f), for some function f (G, γ), and which is either balanced or well correlated with all cuts of conductance at most γ. In a seminal paper, Spielman and Teng γ log 3 V [16] gave an Õ(E/γ2)time algorithm for f = and used it to decompose graphs into a collection of nearexpanders [18]. We present a new spectral algorithm for this problem which runs in time Õ(E/γ) for f = √ γ. Our result yields the first nearlylinear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primaldual semidefiniteprogramming (SDP) approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than γ, somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale’s [3] framework to bound the running time. Our result contains technical ingredients that may be of independent interest.
Flowbased algorithms for local graph clustering
, 2013
"... Given a subset A of vertices of an undirected graph G, the cutimprovement problem asks us to find a subset S that is similar to A but has smaller conductance. An elegant algorithm for this problem has been given by Andersen and Lang [AL08] and requires solving a small number of singlecommodity max ..."
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Given a subset A of vertices of an undirected graph G, the cutimprovement problem asks us to find a subset S that is similar to A but has smaller conductance. An elegant algorithm for this problem has been given by Andersen and Lang [AL08] and requires solving a small number of singlecommodity maximum flow computations over the whole graph G. In this paper, we introduce LocalImprove, the first cutimprovement algorithm that is local, i.e., that runs in time dependent on the size of the input set A rather than on the size of the entire graph. Moreover, LocalImprove achieves this local behavior while closely matching the same theoretical guarantee as the global algorithm of Andersen and Lang. The main application of LocalImprove is to the design of better localgraphpartitioning algorithms. All previously known local algorithms for graph partitioning are randomwalk based and can only guarantee an output conductance of Õ( φopt) when the target set has conductance φopt ∈ [0, 1]. Very recently, Zhu, Lattanzi and Mirrokni [ZLM13] improved this to O(φopt/ Conn) where the internal connectivity parameter Conn ∈ [0, 1] is defined as the reciprocal of the mixing time of the random walk over the graph induced by the target set. This regime is of high practical interest in learning applications as it corresponds to the case when the target set is a wellconnected groundtruth cluster. In this work, we show how to use LocalImprove to obtain a constant approximation O(φopt) as long as Conn/φopt = Ω(1). This yields the first flowbased algorithm for local graph partitioning. Moreover, its performance strictly outperforms the ones based on random walks and surprisingly matches that of the best known global algorithm, which is SDPbased, in this parameter regime [MMV12]. Finally, our results show that spectral methods are not the only viable approach to the construction of local graph partitioning algorithm and open door to the study of algorithms with even better approximation and locality guarantees.