Results 1  10
of
95
Expander Flows, Geometric Embeddings and Graph Partitioning
 IN 36TH ANNUAL SYMPOSIUM ON THE THEORY OF COMPUTING
, 2004
"... We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a ..."
Abstract

Cited by 319 (18 self)
 Add to MetaCart
We give a O( log n)approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)approximation of Leighton and Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.
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 ..."
Abstract

Cited by 242 (14 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 198 (17 self)
 Add to MetaCart
(Show Context)
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
The multiplicative weights update method: a meta algorithm and applications
, 2005
"... Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies ..."
Abstract

Cited by 146 (14 self)
 Add to MetaCart
Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1
A local clustering algorithm for massive graphs and its application to nearlylinear time graph partitioning
, 2013
"... We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal conn ..."
Abstract

Cited by 58 (8 self)
 Add to MetaCart
(Show Context)
We study the design of local algorithms for massive graphs. A local graph algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster—a subset of vertices whose internal connections are significantly richer than its external connections—near a given vertex. The running time of our algorithm, when it finds a nonempty local cluster, is nearly linear in the size of the cluster it outputs. The running time of our algorithm also depends polylogarithmically on the size of the graph and polynomially on the conductance of the cluster it produces. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and webgraphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly linear time algorithm for solving linear systems in symmetric, diagonally dominant matrices. The linear system solver also leads to a nearly linear time algorithm for approximating the secondsmallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.
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 ..."
Abstract

Cited by 36 (6 self)
 Add to MetaCart
(Show Context)
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
A new diffusionbased multilevel algorithm for computing graph partitions of very high quality
 In Proc. 22nd IPDPS
, 2008
"... Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. Highquality partitions are important for many applications, whose objective functions are often NPhard to optimize. Most stateoftheart graph p ..."
Abstract

Cited by 33 (10 self)
 Add to MetaCart
(Show Context)
Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. Highquality partitions are important for many applications, whose objective functions are often NPhard to optimize. Most stateoftheart graph partitioning libraries use a variant of the KernighanLin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called BubbleFOS/C (Meyerhenke et al., IPDPS'06), which optimizes partition shapes by a diffusive mechanism. However, it is too slow for practical use, despite its high solution quality. In this paper, besides proving that BubbleFOS/C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a different diffusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with BubbleFOS/C in a multilevel framework based on two different hierarchy construction methods, we obtain our new graph
Breaking the multicommodity flow barrier for O( √ logn)approximations to sparsest cut
 In FOCS
, 2009
"... This paper ties the line of work on algorithms that find an O ( √ log n)approximation to the sparsest cut together with the line of work on algorithms that run in subquadratic time by using only singlecommodity flows. We present an algorithm that simultaneously achieves both goals, finding an O ..."
Abstract

Cited by 31 (0 self)
 Add to MetaCart
This paper ties the line of work on algorithms that find an O ( √ log n)approximation to the sparsest cut together with the line of work on algorithms that run in subquadratic time by using only singlecommodity flows. We present an algorithm that simultaneously achieves both goals, finding an O ( √ log(n)/ε)approximation using O(n ε log O(1) n) maxflows. The core of the algorithm is a stronger, algorithmic version of Arora et al.’s structure theorem, where we show that matchingchaining argument at the heart of their proof can be viewed as an algorithm that finds good augmenting paths in certain geometric multicommodity flow networks. By using that specialized algorithm in place of a blackbox solver, we are able to solve those instances much more efficiently. We also show the cutmatching game framework can not achieve an approximation any better than Ω(log(n) / log log(n)) without rerouting flow. 1
Dynamics of Large Networks
, 2008
"... A basic premise behind the study of large networks is that interaction leads to complex collective behavior. In our work we found very interesting and counterintuitive patterns for time evolving networks, which change some of the basic assumptions that were made in the past. We then develop models ..."
Abstract

Cited by 30 (0 self)
 Add to MetaCart
A basic premise behind the study of large networks is that interaction leads to complex collective behavior. In our work we found very interesting and counterintuitive patterns for time evolving networks, which change some of the basic assumptions that were made in the past. We then develop models that explain processes which govern the network evolution, fit such models to real networks, and use them to generate realistic graphs or give formal explanations about their properties. In addition, our work has a wide range of applications: it can help us spot anomalous graphs and outliers, forecast future graph structure and run simulations of network evolution. Another important aspect of our research is the study of “local ” patterns and structures of propagation in networks. We aim to identify building blocks of the networks and find the patterns of influence that these blocks have on information or virus propagation over the network. Our recent work included the study of the spread of influence in a large persontoperson