Graph partitioning using single commodity flows

Khandekar, Rohit; Rao, Satish; Vazirani, Umesh

doi:10.1145/1132516.1132574

Cited by 90 publications

(188 citation statements)

References 14 publications

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“…Third, and very recently, there exists an algorithm that uses semidefinite programming to find a solution that is within O( √ log n) of optimal [17]. This paper sparked a flurry of theoretical research on a family of closely related algorithms including [15,99,16], all of which can be informally described as combinations of spectral and flow-based techniques which exploit their complementary strengths. However, none of those algorithms are currently practical enough to use in our study.…”

Section: Approximation Algorithms For Finding Low-conductance Cutsmentioning

confidence: 99%

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,512

1,082

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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 real-world networks, ranging from traditional and on-line social networks, to technological and information networks and web graphs, and ranging in size from thousands up to tens of millions of nodes.Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. Our observations agree with previous work on small networks, but we show that large networks have a very different structure. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales (up to ≈ 100 nodes); and communities of size scale beyond ≈ 100 nodes gradually "blend into" the expanderlike core of the network and thus become less "community-like," with a roughly inverse relationship between community size and optimal community quality. This observation agrees well with the so-called Dunbar number which gives a limit to the size of a well-functioning community.However, this behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. The relatively gradual increase of the network community profile plot as a function of increasing community size depends in a subtle manner on the way in which local clustering information is propagated from smaller to larger size scales in the network. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe i...

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Section: Approximation Algorithms For Finding Low-conductance Cutsmentioning

confidence: 99%

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Leskovec¹,

Lang²,

Dasgupta³

et al. 2009

Internet Mathematics

1,512

1,082

View full text Add to dashboard Cite

show abstract

“…Throughout this paper, we will assume that G is unweighted and simple. A useful application of sparsifiers is in the design of faster algorithms for a range of problems including approximate max-flow [4] and sparsest cut [16]. The idea is that if H is a good approximation of G in an appropriate sense, then it suffices to solve the problem of interest on H rather than on G which would potentially have had many more edges.…”

Section: Introductionmentioning

confidence: 99%

Spectral Sparsification in Dynamic Graph Streams

Ahn

Guha

McGregor

2013

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. We present a new bound relating edge connectivity in a simple, unweighted graph with effective resistance in the corresponding electrical network. The bound is tight. While we believe the bound is of independent interest, our work is motivated by the problem of constructing combinatorial and spectral sparsifiers of a graph, i.e., sparse, weighted sub-graphs that preserve cut information (in the case of combinatorial sparsifiers) and additional spectral information (in the case of spectral sparsifiers). Recent results by Fung et al. (STOC 2011) and Srivastava (SICOMP 2011) show that sampling edges with probability based on edge-connectivity gives rise to a combinatorial sparsifier whereas sampling edges with probability based on effective resistance gives rise to a spectral sparsifier. Our result implies that by simply increasing the sampling probability by a O(n 2/3 ) factor in the combinatorial sparsifier construction, we also preserve the spectral properties of the graph. Combining this with the algorithms of Ahn et al. (SODA 2012, PODS 2012 gives rise to the first data stream algorithm for the construction of spectral sparsifiers in the dynamic setting where edges can be added or removed from the stream. This was posed as an open question by Kelner and Levin (STACS 2011).

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“…In this variant, we have a constraint in the clustering problems that the set of clusters should be disjoint. This constraint has been considered in most of the wellstudied clustering problems in graph theory and combinatorial optimization literature [18,7,25,2,31].…”

Section: Introductionmentioning

confidence: 99%

On the Advantage of Overlapping Clusters for Minimizing Conductance

2013

Self Cite

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Graph clustering is an important problem with applications to bioinformatics, community discovery in social networks, distributed computing, and more. While most of the research in this area has focused on clustering using disjoint clusters, many real datasets have inherently overlapping clusters. We compare overlapping and non-overlapping clusterings in graphs in the context of minimizing their conductance. It is known that allowing clusters to overlap gives better results in practice. We prove that overlapping clustering may be significantly better than non-overlapping clustering with respect to conductance, even in a theoretical setting.For minimizing the maximum conductance over the clusters, we give examples demonstrating that allowing overlaps can yield significantly better clusterings, namely, one that has much smaller optimum. In addition for the min-max variant, the overlapping version admits a simple approximation algorithm, while our algorithm for the non-overlapping version is complex and yields a worse approximation ratio due to the presence of the additional constraint. Somewhat surprisingly, for the problem of minimizing the sum of conductances, we found out that allowing overlap does not help. We show how to apply a general technique to transform any overlapping clustering into a non-overlapping one with only a modest increase in the sum of conductances. This uncrossing technique is of independent interest and may find further applications in the future.We consider this work as a step toward rigorous comparison of overlapping and nonoverlapping clusterings and hope that it stimulates further research in this area. * IBM T.J.Watson research center.

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Graph partitioning using single commodity flows

Cited by 90 publications

References 14 publications

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

Spectral Sparsification in Dynamic Graph Streams

On the Advantage of Overlapping Clusters for Minimizing Conductance

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