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Spectral and Modular Analysis of #P Problems [article]

Ohad Asor
2016 arXiv   pre-print
We present various analytic and number theoretic results concerning the #SAT problem as reflected when reduced into a #PART problem.  ...  As an application we propose a heuristic to probabilistically estimate the solution of #SAT problems.  ...  Acknowledgments Thanks to Avishy Carmi and HunterMinerCrafter for many valuable discussions.  ... 
arXiv:1601.00691v2 fatcat:zf5euhiyifaizez5qm4neptwqa

A vector partitioning approach to detecting community structure in complex networks

Gaoxia Wang, Yi Shen, Ming Ouyang
2008 Computers and Mathematics with Applications  
In recent years, the problem of community structure detection has attracted more and more attention and many approaches have been proposed.  ...  He presents further that this maximization process can be written in terms of the eigenspectrum of the "modularity matrix".  ...  The method combines spectral techniques, a vector partition problem, and the concept of modularity and is a natural extension of bi-partitioning to multiple eigenvectors.  ... 
doi:10.1016/j.camwa.2007.10.028 fatcat:nbe6hy6cwrfbblildcl7ecnlq4

Transitivity based community analysis and detection

Mohammad Aghagolzadeh, Hayder Radha
2013 2013 IEEE Global Conference on Signal and Information Processing  
We propose spectral analysis of the transitivity gradient matrix and compare our framework to the modularity based community detection that attracted many network researchers' attention recently.  ...  This paper extends our previous effort in employing transitivity attributes of graphs for social network analysis. Specifically, here we focus on the problem of network community detection.  ...  Modularity matrix B = A − P where [P ] ij = d i d j /(2m) with d idenoting the degree of node i and m the size of the graph.  ... 
doi:10.1109/globalsip.2013.6736909 dblp:conf/globalsip/AghagolzadehR13 fatcat:trrpvfzpojbg5fmcepl6u343pq

Co-clustering for Binary and Categorical Data with Maximum Modularity

Lazhar Labiod, Mohamed Nadif
2011 2011 IEEE 11th International Conference on Data Mining  
Second, the maximization of the extended modularity is shown as a trace maximization problem.  ...  In this paper we propose a spectral based clustering algorithm to maximize an extended Modularity measure for categorical data; first, we establish the connection with the Relational Analysis criterion  ...  and let P = M m=1 p m denote the full number of categories of all variables.  ... 
doi:10.1109/icdm.2011.37 dblp:conf/icdm/LabiodN11 fatcat:nkbyklrzvnbgxo5jzrln5kpn7e

Detectability of the spectral method for sparse graph partitioning

T. Kawamoto, Y. Kabashima
2015 Europhysics letters  
We show that modularity maximization with the resolution parameter offers a unifying framework of graph partitioning.  ...  In this framework, we demonstrate that the spectral method exhibits universal detectability, irrespective of the value of the resolution parameter, as long as the graph is partitioned.  ...  This work was supported by JSPS KAKENHI No. 26011023 (TK) and No. 25120013 (YK) and the JSPS Core-to-Core Program "Non-equilibrium dynamics of soft matter and information".  ... 
doi:10.1209/0295-5075/112/40007 fatcat:r24wjdf2ejb5dl2fhl76joy2pu

A Perspective Analysis of Hidden Community Mining Methods in Large Scale Social Networks

Renuga Devi.R, Hemalatha. M
2013 International Journal of Computer Applications  
Here we analysis the various community mining techniques which is already available. Such as MinCut algorithm, Regression based algorithm, Max-Min modularity measure, LM algorithm and SECI model.  ...  Research in social network analysis has increased in recent years. Because of the popularity of the social networking sites, many researchers concentrate on this area for research.  ...  The spectral clustering methods [14, 15, and 16] and modularity based algorithms [17, 18]. However, none of them differentiate the essential features of the area of the network in question.  ... 
doi:10.5120/13089-0368 fatcat:zhgvbif2kfcx7ohtdqnnogctne

Diffusion Model Based Spectral Clustering for Protein-Protein Interaction Networks

Kentaro Inoue, Weijiang Li, Hiroyuki Kurata, Ingemar T. Ernberg
2010 PLoS ONE  
Methodology/Principal Findings: To address this problem, we propose a diffusion model-based spectral clustering algorithm, which analytically solves the cluster structure of PPI networks as a problem of  ...  However, clear modular decomposition is often hard due to the heterogeneous or scale-free properties of PPI networks.  ...  The maximization of modularity is one of widely-used criteria and can be reformulated as an eigenvector problem of the spectral clustering algorithm [33, 41] .  ... 
doi:10.1371/journal.pone.0012623 pmid:20830307 pmcid:PMC2935381 fatcat:vglkt5d5l5fblkcars6tiqolmm

A Study on Relationship between Modularity and Diffusion Dynamics in Networks from Spectral Analysis Perspective

Kiyotaka Ide, Akira Namatame, Loganathan Ponnambalam, Fu Xiuju, Rick Siow
2014 International Journal of Advanced Computer Science and Applications  
In this paper, we analysed spectral properties of the networks and diffusion dynamics. Especially, we focus on studying the relationship between modularity and diffusion dynamics.  ...  Our analysis as well as simulation results show that the relative influences from the non-largest eigenvalues and the corresponding eigenvectors increase when modularity of network increases.  ...  However, according to the results of our analysis from spectral point of view and numerical simulations, the accuracy of this approximation method varies depending on the modularity of the network.  ... 
doi:10.14569/ijacsa.2014.050905 fatcat:xcpm2w3grnfgnebgbrw6io2dk4

Comparative analysis on the selection of number of clusters in community detection

Tatsuro Kawamoto, Yoshiyuki Kabashima
2018 Physical review. E  
From the analysis, the tendency of overfit and underfit that the assessment criteria and algorithms have, becomes apparent.  ...  In this paper we focus on the framework based on a stochastic block model, and investigate the performance of greedy algorithms, statistical inference, and spectral methods.  ...  Generating the stochastic block model is a forward problem and its inverse problem, that is, the inference of γ, ω, and σ, must be done for community detection. B.  ... 
doi:10.1103/physreve.97.022315 pmid:29548181 fatcat:xeyf2sk5jze7xgjojqvw5dke3u

The Missing Spectral Basis in Algebra and Number Theory

Garret Sobczyk
2001 The American mathematical monthly  
ACKNOWLEDGMENT The author gratefully acknowledges support given by INIP of the Universidad de las Américas-Puebla.  ...  Multiplying both sides of (4) by x gives The modular number systems Z p m , modulo a power of a prime, play a particularly important role in number theory because all modular problems reduce to problems  ...  We have b −1 = r i=1 (b i(m i −1) · · · b i0 ) −1 p i s i , so the problem of finding b −1 ∈ Z h is reduced to the problem of finding the inverse in each of the prime power modular number systems Z p m  ... 
doi:10.1080/00029890.2001.11919757 fatcat:w7wmbjfbjvde3i2jrudkh4fddu

The Missing Spectral Basis in Algebra and Number Theory

Garret Sobczyk
2001 The American mathematical monthly  
ACKNOWLEDGMENT The author gratefully acknowledges support given by INIP of the Universidad de las Américas-Puebla.  ...  Multiplying both sides of (4) by x gives The modular number systems Z p m , modulo a power of a prime, play a particularly important role in number theory because all modular problems reduce to problems  ...  We have b −1 = r i=1 (b i(m i −1) · · · b i0 ) −1 p i s i , so the problem of finding b −1 ∈ Z h is reduced to the problem of finding the inverse in each of the prime power modular number systems Z p m  ... 
doi:10.2307/2695240 fatcat:ryzxq5c3drgife5nwasf3vjgmq

Kernel spectral clustering for community detection in complex networks

Rocco Langone, Carlos Alzate, Johan A. K. Suykens
2012 The 2012 International Joint Conference on Neural Networks (IJCNN)  
We show the use of kernel spectral clustering for the analysis of unweighted networks. We employ the primaldual framework and make use of out-of-sample extensions.  ...  We demonstrate the effectiveness of our model on synthetic networks and benchmark data from real networks (power grid network and protein interaction network of yeast).  ...  The corresponding eigenvalue problem becomes P r = ξr, and as we show afterward it can be viewed as the dual problem of a constrained optimization problem typical of least squares support vector machines  ... 
doi:10.1109/ijcnn.2012.6252726 dblp:conf/ijcnn/LangoneAS12 fatcat:vtzvesp3nvetppawtxcrrntqpm

Comparative study for inference of hidden classes in stochastic block models

Pan Zhang, Florent Krzakala, Jörg Reichardt, Lenka Zdeborová
2012 Journal of Statistical Mechanics: Theory and Experiment  
Inference of hidden classes in stochastic block model is a classical problem with important applications.  ...  We show that belief propagation shows much better performance when compared to naïve mean field and spectral approaches.  ...  JR was supported by a Fellowship Computational Sciences of the Volkswagen Foundation.  ... 
doi:10.1088/1742-5468/2012/12/p12021 fatcat:7uvcz7mrybgxlblz45fwbqvffa

Geometric multiscale community detection: Markov stability and vector partitioning

Zijing Liu, Mauricio Barahona
2017 Journal of Complex Networks  
We apply the algorithm to the spectral optimisation of modularity and Markov Stability community detection.  ...  The spectral embedding based on the transition matrix eigenvectors leads to improved partitions with higher information content and higher modularity than the eigen-decomposition of the modularity matrix  ...  Acknowledgements The authors would like to thank Dr Michael Schaub for extended discussions, and Dr Sam Greenbury for reading the manuscript and giving useful comments.  ... 
doi:10.1093/comnet/cnx028 fatcat:njxpa3ed6nbu5j2wpojvynjktq

Spectral methods for the detection of network community structure: a comparative analysis

Hua-Wei Shen, Xue-Qi Cheng
2010 Journal of Statistical Mechanics: Theory and Experiment  
matrix, the modularity matrix, the correlation matrix and several other variants of these matrices.  ...  This indicates that it is crucial to take into account the heterogeneous distribution of node degree when using spectral analysis for the detection of community structure.  ...  The authors also thank J M Huang, P Du, X F Zhu, and P Cao for useful discussions and suggestions.  ... 
doi:10.1088/1742-5468/2010/10/p10020 fatcat:o22tqwr6nvdjhg23pw2tufuspm
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