Expander flows, geometric embeddings and graph partitioning

Arora, Sanjeev; Rao, Satish; Vazirani, Umesh

doi:10.1145/1502793.1502794

Cited by 329 publications

(224 citation statements)

References 39 publications

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“…The conductance of a graph G is the minimum conductance of any subset of nodes:

ϕ false(G false) = min_{S \subset V} ϕ false(S false) .

Computing the conductance ϕ( G ) of an arbitrary graph is an intractable problem (in the sense that the associated decision problem is NP-hard [71]), but this quantity can be approximated by the second smallest eigenvalue λ 2 of the normalized Laplacian [69, 70]. …”

Section: Network Community Profiles (Ncps) and Their Interpretationmentioning

confidence: 99%

Think locally, act locally: Detection of small, medium-sized, and large communities in large networks

et al. 2015

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It is common in the study of networks to investigate intermediate-sized (or “meso-scale”) features to try to gain an understanding of network structure and function. For example, numerous algorithms have been developed to try to identify “communities,” which are typically construed as sets of nodes with denser connections internally than with the remainder of a network. In this paper, we adopt a complementary perspective that “communities” are associated with bottlenecks of locally-biased dynamical processes that begin at seed sets of nodes, and we employ several different community-identification procedures (using diffusion-based and geodesic-based dynamics) to investigate community quality as a function of community size. Using several empirical and synthetic networks, we identify several distinct scenarios for “size-resolved community structure” that can arise in real (and realistic) networks: (i) the best small groups of nodes can be better than the best large groups (for a given formulation of the idea of a good community); (ii) the best small groups can have a quality that is comparable to the best medium-sized and large groups; and (iii) the best small groups of nodes can be worse than the best large groups. As we discuss in detail, which of these three cases holds for a given network can make an enormous difference when investigating and making claims about network community structure, and it is important to take this into account to obtain reliable downstream conclusions. Depending on which scenario holds, one may or may not be able to successfully identify “good” communities in a given network (and good communities might not even exist for a given community quality measure), the manner in which different small communities fit together to form meso-scale network structures can be very different, and processes such as viral propagation and information diffusion can exhibit very different dynamics. In addition, our results suggest that, for many large realistic networks, the output of locally-biased methods that focus on communities that are centered around a given seed node might have better conceptual grounding and greater practical utility than the output of global community-detection methods. They also illustrate subtler structural properties that are important to consider in the development of better benchmark networks to test methods for community detection.

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“…The conductance of a graph G is the minimum conductance of any subset of nodes:

ϕ false(G false) = min_{S \subset V} ϕ false(S false) .

Section: Network Community Profiles (Ncps) and Their Interpretationmentioning

confidence: 99%

Think locally, act locally: Detection of small, medium-sized, and large communities in large networks

et al. 2015

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“…This means that the objective function should have a few edges crossing the cut, and the cut should be close to the bisection. The sparsest cut objective for clustering can be defined as (Arora et al, 2009)…”

Section: Visualization Of Clustering Process Using Graph Theorymentioning

confidence: 99%

Correlation clustering methodologies and their fundamental results

2017

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Correlation clustering possibly represents the most intuitive form of clustering construction. It gives solutions that can be approximated while automatically selecting the number of clusters. This approach handles scenarios where the focus is on relationships between the objects instead of on actual representations of the objects. The suitability of this method extends to the structured objects, for which feature vectors are not easy to obtain. Given the increasing scale of data these days, correlation clustering has become a powerful addition to the fields of data mining and agnostic learning. Correlation clustering considers a weighted graph G=(V,E), where the edge weight indicates whether two nodes are similar (positive edge weight) or different (negative edge weight). The task is to find a clustering that either maximizes agreements or minimizes disagreements. Unlike other clustering algorithms, this does not require choosing the number of clusters (k) in advance. The objective to minimize the sum of weights of the cut edges is independent of the number of clusters. Methodologies, such as approximations and linear programming formulations, have been used to approach this problem. This paper focuses on the problem of correlation clustering and lists the solutions proposed by various researchers. These solutions approach the problem using different computational techniques. Correlation clustering‐based applications such as entity de‐duplication, signed social networks, and problem of aggregating multiples have also been discussed.

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“…In the second step of our algorithm, we also use the following theorem by Arora, Lee, and Naor [3] (see also [4]). …”

Section: For All Non-zero Vectorsū Andv ϕ(ū)mentioning

confidence: 99%