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Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices
"... Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Lapla ..."
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Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including lowstretch spanning trees, graph sparsifiers, ultrasparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective.
Fast Approximation Algorithms for Cutbased Problems in Undirected Graphs
"... We present a general method of designing fast approximation algorithms for cutbased minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs wh ..."
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Cited by 19 (3 self)
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We present a general method of designing fast approximation algorithms for cutbased minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs while only losing a polylogarithmic factor in the approximation guarantee. To illustrate the applicability of our paradigm, we focus our attention on the undirected sparsest cut problem with general demands and the balanced separator problem. By a simple use of our framework, we obtain polylogarithmic approximation algorithms for these problems that run in time close to linear. The main tool behind our result is an efficient procedure that decomposes general graphs into simpler ones while approximately preserving the cutflow structure. This decomposition is inspired by the cutbased graph decomposition of Räcke that was developed in the context of oblivious routing schemes, as well as, by the construction of the ultrasparsifiers due to Spielman and Teng that was employed to preconditioning symmetric diagonallydominant matrices. 1
Divide and Conquer: Partitioning Online Social Networks
, 2009
"... Online Social Networks (OSNs) have exploded in terms of scale and scope over the last few years. The unprecedented growth of these networks present challenges in terms of system design and maintenance. One way to cope with this is by partitioning such large networks and assigning these partitions to ..."
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Cited by 14 (5 self)
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Online Social Networks (OSNs) have exploded in terms of scale and scope over the last few years. The unprecedented growth of these networks present challenges in terms of system design and maintenance. One way to cope with this is by partitioning such large networks and assigning these partitions to different machines. However, social networks possess unique properties that make the partitioning problem nontrivial. The main contribution of this paper is to understand different properties of social networks and how these properties can guide the choice of a partitioning algorithm. Using large scale measurements representing real OSNs, we first characterize different properties of social networks, and then we evaluate qualitatively different partitioning methods that cover the design space. We expose different tradeoffs involved and understand them in light of properties of social networks. We show that a judicious choice of a partitioning scheme can help improve performance. 1.
Learning with Partially Absorbing Random Walks
 In Proceedings of Annual Conference on Neural Information Processing Systems (NIPS12
, 2012
"... We propose a novel stochastic process that is with probability αi being absorbed at current state i, and with probability 1 − αi follows a random edge out of it. We analyze its properties and show its potential for exploring graph structures. We prove that under proper absorption rates, a random wal ..."
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We propose a novel stochastic process that is with probability αi being absorbed at current state i, and with probability 1 − αi follows a random edge out of it. We analyze its properties and show its potential for exploring graph structures. We prove that under proper absorption rates, a random walk starting from a set S of low conductance will be mostly absorbed in S. Moreover, the absorption probabilities vary slowly inside S, while dropping sharply outside, thus implementing the desirable cluster assumption for graphbased learning. Remarkably, the partially absorbing process unifies many popular models arising in a variety of contexts, provides new insights into them, and makes it possible for transferring findings from one paradigm to another. Simulation results demonstrate its promising applications in retrieval and classification. 1
SPARSE QUADRATIC FORMS AND THEIR GEOMETRIC APPLICATIONS
"... In what follows all matrices are assumed to have real entries, and square matrices are always assumed to be symmetric unless stated otherwise. The support of a k × n matrix A = (aij) will be denoted below by supp(A) = { (i, j) ∈ {1,..., k} × {1,..., n} : aij ̸ = 0}. ..."
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Cited by 11 (0 self)
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In what follows all matrices are assumed to have real entries, and square matrices are always assumed to be symmetric unless stated otherwise. The support of a k × n matrix A = (aij) will be denoted below by supp(A) = { (i, j) ∈ {1,..., k} × {1,..., n} : aij ̸ = 0}.
Finding Endogenously Formed Communities
, 2012
"... A central problem in data mining and social network analysis is determining overlapping communities (clusters) among individuals or objects in the absence of external identification or tagging. We address this problem by introducing a framework that captures the notion of communities or clusters det ..."
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A central problem in data mining and social network analysis is determining overlapping communities (clusters) among individuals or objects in the absence of external identification or tagging. We address this problem by introducing a framework that captures the notion of communities or clusters determined by the relative affinities among their members. To this end we define what we call an affinity system, which is a set of elements, each with a vector characterizing its preference for all other elements in the set. We define a natural notion of (potentially overlapping) communities in an affinity system, in which the members of a given community collectively prefer each other to anyone else outside the community. Thus these communities are endogenously formed in the affinity system and are “selfdetermined ” or
Approximating the exponential, the Lanczos method, and an Õ(m)time spectral algorithm for balanced separator
 IN: PROCEEDINGS OF THE 44TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2012
"... We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O ( √ γ) in G, or outputs a certificate that all bbalanced cuts in G have conductance at le ..."
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Cited by 10 (3 self)
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We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O ( √ γ) in G, or outputs a certificate that all bbalanced cuts in G have conductance at least γ, and runs in time Õ(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(−L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrixexponentialvector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(−A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector u, in time roughly Õ(m A), where m A is the number of nonzero entries of A. This uses, in a nontrivial way, the breakthrough result of Spielman and Teng on inverting symmetric and diagonallydominant matrices in Õ(m A) time. Finally, we prove that e −x can be uniformly approximated up to a small additive error, in a nonnegative interval [a, b] with a polynomial of
Scalable Discovery of Best Clusters on Large Graphs
"... The identification of clusters, wellconnected components in a graph, is useful in many applications from biological function prediction to social community detection. However, finding these clusters can be difficult as graph sizes increase. Most current graph clustering algorithms scale poorly in t ..."
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Cited by 10 (1 self)
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The identification of clusters, wellconnected components in a graph, is useful in many applications from biological function prediction to social community detection. However, finding these clusters can be difficult as graph sizes increase. Most current graph clustering algorithms scale poorly in terms of time or memory. An important insight is that many clustering applications need only the subset of best clusters, and not all clusters in the entire graph. In this paper we propose a new technique, Top Graph Clusters (TopGC), which probabilistically searches large, edge weighted, directed graphs for their best clusters in linear time. The algorithm is inherently parallelizable, and is able to find variable size, overlapping clusters. To increase scalability, a parameter is introduced that controls memory use. When compared with three other stateofthe art clustering techniques, TopGC achieves running time speedups of up to 70% on large scale real world datasets. In addition, the clusters returned by TopGC are consistently found to be better both in calculated score and when compared on real world benchmarks. 1.