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Statistical properties of community structure in large social and information networks
"... A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structur ..."
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Cited by 242 (14 self)
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A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales, and we study over 70 large sparse realworld networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large realworld networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually “blend in ” with the rest of the network and thus become less “communitylike.” This behavior is not explained, even at a qualitative level, by any of the commonlyused network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are wellembeddable in a lowdimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community structure similar to our observations.
Community structure in large networks: Natural cluster sizes and the absence of large welldefined clusters
, 2008
"... A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins wit ..."
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Cited by 200 (17 self)
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A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a “real ” communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales. We study over 100 large realworld networks, ranging from traditional and online social networks, to technological and information networks and
Efficient Algorithms for Maximum Lifetime Data Gathering and Aggregation in Wireless Sensor Networks
, 2002
"... The rapid advances in processor, memory, and radio technology have enabled the development of distributed networks of small, inexpensive nodes that are capable of sensing, computation, and wireless communication. Sensor networks of the future are envisioned to revolutionize the paradigm of collect ..."
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Cited by 96 (4 self)
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The rapid advances in processor, memory, and radio technology have enabled the development of distributed networks of small, inexpensive nodes that are capable of sensing, computation, and wireless communication. Sensor networks of the future are envisioned to revolutionize the paradigm of collecting and processing information in diverse environments. However, the severe energy constraints and limited computing resources of the sensors, present major challenges for such a vision to become a reality. We consider
A Combinatorial, PrimalDual approach to Semidefinite Programs
"... Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primaldual approach to solve SDPs using a generalization of the wellknown multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced ..."
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Cited by 93 (10 self)
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Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primaldual approach to solve SDPs using a generalization of the wellknown multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced Separator in undirected and directed weighted graphs, and the Min UnCut problem, this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primaldual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones.
Beyond pairwise clustering
 in IEEE Computer Society Conference on Computer Vision and Pattern Recognition
"... We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a twostep algorithm for solving this problem. In the first step we use a nove ..."
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Cited by 61 (3 self)
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We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a twostep algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms. 1.
Improved approximation algorithms for unsplittable flow problems (Extended Abstract)
 IN PROCEEDINGS OF THE 38TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 1997
"... In the singlesource unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single st i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by it ..."
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Cited by 44 (2 self)
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In the singlesource unsplittable flow problem we are given a graph G; a source vertex s and a set of sinks t 1 ; : : : ; t k with associated demands. We seek a single st i flow path for each commodity i so that the demands are satisfied and the total flow routed across any edge e is bounded by its capacity c e : The problem is an NPhard variant of max flow and a generalization of singlesource edgedisjoint paths with applications to scheduling, load balancing and virtualcircuit routing problems. In a significant development, Kleinberg gave recently constantfactor approximation algorithms for several natural optimization versions of the problem [18]. In this paper we give a generic framework that yields simpler algorithms and significant improvements upon the constant factors. Our framework, with appropriate subroutines, applies to all optimization versions previously considered and treats in a unified manner directed and u...
Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs
, 2010
"... We introduce a new approach to computing an approximately maximum st flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be ..."
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Cited by 40 (5 self)
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We introduce a new approach to computing an approximately maximum st flow in a capacitated, undirected graph. This flow is computed by solving a sequence of electrical flow problems. Each electrical flow is given by the solution of a system of linear equations in a Laplacian matrix, and thus may be approximately computed in nearlylinear time. Using this approach, we develop the fastest known algorithm for computing approximately maximum st flows. For a graph having n vertices and m edges, our algorithm computes a (1−ɛ)approximately maximum st flow in time 1 Õ ( mn 1/3 ɛ −11/3). A dual version of our approach computes a (1 + ɛ)approximately minimum st cut in time Õ ( m + n 4/3 ɛ −16/3) , which is the fastest known algorithm for this problem as well. Previously, the best dependence on m and n was achieved by the algorithm of Goldberg and Rao (J. ACM 1998), which can be used to compute approximately maximum st flows in time Õ ( m √ nɛ −1) , and approximately minimum st cuts in time Õ ( m + n 3/2 ɛ −3). Research partially supported by NSF grant CCF0843915.
The Pseudoflow algorithm: A new algorithm for the maximum flow problem
 Operations Research
, 2008
"... We introduce the pseudoflow algorithm for the maximumflow problem that employs only pseudoflows and does not generate flows explicitly. The algorithm solves directly a problem equivalent to the minimumcut problem—the maximum blockingcut problem. Once the maximum blockingcut solution is available ..."
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Cited by 37 (12 self)
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We introduce the pseudoflow algorithm for the maximumflow problem that employs only pseudoflows and does not generate flows explicitly. The algorithm solves directly a problem equivalent to the minimumcut problem—the maximum blockingcut problem. Once the maximum blockingcut solution is available, the additional complexity required to find the respective maximumflow is O�mlog n�. A variant of the algorithm is a new parametric maximumflow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximumflow problem. The pseudoflow algorithm has also a simplex variant, pseudoflowsimplex, that can be implemented to solve the maximumflow problem. One feature of the pseudoflow algorithm is that it can initialize with any pseudoflow. This feature allows it to reach an optimal solution quickly when the initial pseudoflow is “close ” to an optimal solution. The complexities of the pseudoflow algorithm, the pseudoflowsimplex, and the parametric variants of pseudoflow and pseudoflowsimplex algorithms are all O�mnlog n � on a graph with n nodes and m arcs. Therefore, the pseudoflowsimplex algorithm is the fastest simplex algorithm known for the parametric maximumflow problem. The pseudoflow algorithm is also shown to solve the maximumflow problem on s�ttree networks in linear time, where s�ttree networks are formed by joining a forest of capacitated arcs, with nodes s and t adjacent to any subset of the nodes. Subject classifications: flow algorithms; parametric flow; normalized tree; lowest label; pseudoflow algorithm; maximum flow.
On partitioning graphs via single commodity flows
 In STOC ’08: Proceedings of the 40th Annual ACM Symposium on Theory of Computing
, 2008
"... In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cutmatching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approxima ..."
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Cited by 34 (6 self)
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In this paper we obtain improved upper and lower bounds for the best approximation factor for Sparsest Cut achievable in the cutmatching game framework proposed in Khandekar et al. [9]. We show that this simple framework can be used to design combinatorial algorithms that achieve O(log n) approximation factor and whose running time is dominated by a polylogarithmic number of singlecommodity maxflow computations. This matches the performance of the algorithm of Arora and Kale [2]. Moreover, we also show that it is impossible to get an approximation factor of better than Ω ( √ log n) in the cutmatching game framework. These results suggest that the simple and concrete abstraction of the cutmatching game may be powerful enough to capture the essential features of the complexity of Sparsest Cut. Categories and Subject Descriptors