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Convex Relaxation Methods for Community Detection [article]

Xiaodong Li, Yudong Chen, Jiaming Xu
2018 arXiv   pre-print
This paper surveys recent theoretical advances in convex optimization approaches for community detection.  ...  We introduce some important theoretical techniques and results for establishing the consistency of convex community detection under various statistical models.  ...  implementations of these approaches for practical and large-scale problems.  ... 
arXiv:1810.00315v1 fatcat:6fk7op7iqvcq3h3aspsngv5e3y

Post-Processing Partitions to Identify Domains of Modularity Optimization

William Weir, Scott Emmons, Ryan Gibson, Dane Taylor, Peter Mucha
2017 Algorithms  
Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition ---i.e., the parameter-space domain where it has the largest modularity relative to the input set---  ...  We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational  ...  The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.  ... 
doi:10.3390/a10030093 pmid:29046743 pmcid:PMC5642987 fatcat:u7fen6fjezd63k5bu6cuc4oox4

Anomalous subgraph detection via Sparse Principal Component Analysis

Navraj Singh, Benjamin A. Miller, Nadya T. Bliss, Patrick J. Wolfe
2011 2011 IEEE Statistical Signal Processing Workshop (SSP)  
We characterize the anomaly in a subgraph via the well-known notion of network modularity, and we show that the optimization problem formulation resulting from our setup is very similar to a recently introduced  ...  In this paper we investigate a problem with applicability to a wide variety of domains -detecting small, anomalous subgraphs in a background graph.  ...  The reader is also referred to [11] for some previous related work on mathematical programming formulations for the related problem of modularity maximization.  ... 
doi:10.1109/ssp.2011.5967738 fatcat:lkmrxmp2zzewdpahgjk3jzrjzm

Detecting communities using asymptotical surprise

V. A. Traag, R. Aldecoa, J.-C. Delvenne
2015 Physical Review E  
Methods to detect these groups of nodes usually maximize an objective function, which implicitly contains the definition of a community.  ...  This allows for the development of an efficient algorithm for optimizing surprise. Incidentally, this leads to a straightforward extension of surprise to weighted graphs.  ...  Finally, We show the good performance of Surprise maximization for community detection, especially in large graphs with small communities.  ... 
doi:10.1103/physreve.92.022816 pmid:26382463 fatcat:rvedpcpm3nb67cllvy2ofuerw4

Identification of community structure in networks with convex optimization [article]

Roland Hildebrand
2008 arXiv   pre-print
We reformulate the problem of modularity maximization over the set of partitions of a network as a conic optimization problem over the completely positive cone, converting it from a combinatorial optimization  ...  We use our method to provide the first proof of optimality of a partition for a real-world network.  ...  The author would like to thank the members of this project for this possibility. This paper presents research results of the project CARESSE of the pole MSTIC of Université Joseph Fourier, France.  ... 
arXiv:0806.1896v2 fatcat:vkl4jcdejjhvxmju4qgx5yr2w4

Modularity and Mutual Information in Networks: Two Sides of the Same Coin [article]

Qian Wang, Yongkang Guo, Zhihuan Huang, Yuqing Kong
2022 arXiv   pre-print
Previous work on modularity has developed many efficient algorithms for modularity maximization. However, few of researchers considered the interpretation of the modularity function itself.  ...  It can serve as both a standard benchmark to compare different community detection algorithms, and an optimization objective to detect communities itself.  ...  Related Work Modularity was first proposed by [Newman and Girvan, 2004] as a stop criterion for another community detection algorithm.  ... 
arXiv:2103.02542v2 fatcat:ildimuoahbhqxhkjtz6wkk74gi

Additive Approximation Algorithms for Modularity Maximization [article]

Yasushi Kawase, Tomomi Matsui, Atsushi Miyauchi
2016 arXiv   pre-print
Community detection in graphs is now often conducted through modularity maximization: given an undirected graph G=(V,E), we are asked to find a partition C of V that maximizes the modularity.  ...  The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004).  ...  Quality functions in community detection return some value that represents the community-degree for a given partition of the set of vertices.  ... 
arXiv:1601.03316v1 fatcat:w2nqir2fqfdbbkiismxeuvi4hq

Additive Approximation Algorithms for Modularity Maximization

Yasushi Kawase, Tomomi Matsui, Atsushi Miyauchi, Marc Herbstritt
2016 International Symposium on Algorithms and Computation  
Community detection in graphs is now often conducted through modularity maximization: given an undirected graph G = (V, E), we are asked to find a partition C of V that maximizes the modularity.  ...  The modularity is a quality function in community detection, which was introduced by Newman and Girvan (2004) .  ...  Quality functions in community detection return some value that represents the community-degree for a given partition of the set of vertices.  ... 
doi:10.4230/lipics.isaac.2016.43 dblp:conf/isaac/KawaseMM16 fatcat:dsq4yugxrzethbsuzc5rplyuwq

Applications of reformulations in mathematical programming

Alberto Costa
2012 4OR  
The third application considered in the thesis is that of computing the convex relaxation for multilinear problems, and to compare the "primal" formulation and another one obtained using a "dual" representation  ...  The first problem tackled herein is graph clustering by means of modularity maximization. Since this problem is NP-hard, several heuristics are proposed.  ...  However, modularity maximization remains a very interesting criterion for the detection of clusters; for the goal of this thesis, one of its most interesting properties is the fact that it can be described  ... 
doi:10.1007/s10288-012-0220-1 fatcat:47jmhf5adfg73joxngfmrsp44e

Quantitative function for community detection

Zhenping Li, Shihua Zhang, Rui-Sheng Wang, Xiang-Sun Zhang, Luonan Chen
2008 Physical Review E  
We propose a quantitative function for community partition-i.e., modularity density or D value.  ...  communities.  ...  So the community-detection problem can be viewed as a problem of finding a partition of a network such that its modularity density D is maximized.  ... 
doi:10.1103/physreve.77.036109 pmid:18517463 fatcat:rasvd7ixvzcatjsvlkgpbdbecq

Convexified Modularity Maximization for Degree-corrected Stochastic Block Models [article]

Yudong Chen and Xiaodong Li and Jiaming Xu
2016 arXiv   pre-print
Our approach is based on a convex programming relaxation of the classical (generalized) modularity maximization formulation, followed by a novel doubly-weighted ℓ_1 -norm k -median procedure.  ...  This paper proposes a convexified modularity maximization approach for estimating the hidden communities under DCSBM.  ...  Appendices A Supporting lemmas In this section we state several additional technical lemmas concerning random matrices. These lemmas are used in the proof of our main theorems.  ... 
arXiv:1512.08425v2 fatcat:kczixcdhejesznn5v2spzhypoe

Simplified Energy Landscape for Modularity Using Total Variation

Zachary M. Boyd, Egil Bae, Xue-Cheng Tai, Andrea L. Bertozzi
2018 SIAM Journal on Applied Mathematics  
We show that modularity optimization is equivalent to minimizing a convex TV-based functional over a discrete domain---again, assuming the number of communities is known.  ...  Furthermore, we show that modularity has no convex relaxation satisfying certain natural conditions.  ...  Modularity also has a preferred scale for communities [25, 46] .  ... 
doi:10.1137/17m1138972 fatcat:odng4qt3vvcmxlf3z2bpjfzweq

Simplified Energy Landscape for Modularity Using Total Variation [article]

Zachary Boyd, Egil Bae, Xue-Cheng Tai, Andrea L. Bertozzi
2018 arXiv   pre-print
We show that modularity optimization is equivalent to minimizing a convex TV-based functional over a discrete domain, again, assuming the number of communities is known.  ...  They solve this problem, assuming the number of communities is known, using a Merriman, Bence, Osher (MBO) scheme.  ...  modularity, and • any convergent subsequence of the u converges to a maximizer of modularity.  ... 
arXiv:1707.09285v3 fatcat:mv66timsonh4tlxtlj352yybwq

An ADMM Algorithm for Clustering Partially Observed Networks [article]

Necdet Serhat Aybat, Sahar Zarmehri, Soundar Kumara
2015 arXiv   pre-print
In order to detect the underlying cluster structure, we propose a new convex formulation to decompose a partially observed adjacency matrix of a network into low-rank and sparse components.  ...  Many of these algorithms are based on modularity maximization, and these methods suffer from the resolution limit.  ...  for community detection.  ... 
arXiv:1410.3898v2 fatcat:vmdp6cnitrbajmkhrolhtxgjvy

Botnet Detection using Social Graph Analysis [article]

Jing Wang, Ioannis Ch. Paschalidis
2015 arXiv   pre-print
The latter stage uses a refined modularity measure and formulates the problem as a non-convex optimization problem for which appropriate relaxation strategies are developed.  ...  The method consists of two stages: (i) anomaly detection in an "interaction" graph among nodes using large deviations results on the degree distribution, and (ii) community detection in a social "correlation  ...  Our criterion for "appropriate" is related to the well-known concept of modularity in community detection [17] , [18] , [19] . 2) Modularity-based Community Detection: The problem of community detection  ... 
arXiv:1503.02337v1 fatcat:2bujjptmanekhkjesndwiy6vza
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