CoAssociation MatrixBased MultiLayer Fusion for Community Detection in Attributed Networks
Abstract
:1. Introduction
 We propose a twolayer representation for attributed networks. In the view of the representation, the lowerlayer representation is the raw data from network topological structure and node attributes, respectively, while the higherlayer representation is a set of community partitions that are generated from lowerlayer data by using the existing community detection algorithms.
 We propose a weighted coassociation matrixbased community ensemble method for community detection in attributed networks. In order to reduce the uncertainty and data inconsistency, the WCMFA employs the coassociation matrix to learn optimum community structure with the twolayer representation of raw data.
 We also empirically evaluate the effectiveness of WCMFA. The experiment results show that our proposed community detection algorithm, WCMFA, is the optimal choice to detect community structure in attributed networks.
2. Related Work
3. Proposed Method
3.1. Notation and Method Overview
 ${P}^{\ast}$ should be robustness to small variations in $\mathcal{P}$, according to the fusion strategy;
 ${P}^{\ast}$ should have better performance than each candidate community partition $P\in \mathcal{P}$, statistically;
 ${P}^{\ast}$ should be similar to all single community partition.
3.2. TwoLayer Representation and Community Fusion
3.3. Community Fusion Algorithm
Algorithm 1 The weighted coassociation matrix based community fusion algorithm, WCMFA, which detects community structure in attributed networks based on the twolayer representation 

4. Experiments
4.1. Experiment Setup
4.2. Results
4.3. Impact of Varying Size of Candidate Community Partitions and Nodes
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
 Rubinov, M.; Sporns, O. Complex network measures of brain connectivity: Uses and interpretations. Neuroimage 2010, 52, 1059–1069. [Google Scholar] [CrossRef] [PubMed]
 Han, J.; Li, W.; Deng, W. Multiresolution community detection in massive networks. Sci. Rep. 2016, 6, 38998. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Fortunato, S.; Hric, D. Community detection in networks: A user guide. Phys. Rep. 2016, 659, 1–44. [Google Scholar] [CrossRef][Green Version]
 Kernighan, B.W.; Lin, S. An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 1970, 49, 291–307. [Google Scholar] [CrossRef]
 Girvan, M.; Newman, M.E.J. Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 2002, 99, 7821–7826. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Newman, M.E.J.; Girvan, M. Finding and evaluating community structure in networks. Phys. Rev. E 2004, 69, 026113. [Google Scholar] [CrossRef]
 Raghavan, U.N.; Albert, R.; Kumara, S. Near linear time algorithm to detect community structures in largescale networks. Phys. Rev. E 2007, 76, 036106. [Google Scholar] [CrossRef]
 Blondel, V.D.; Guillaume, J.L.; Lambiotte, R.; Lefebvre, E. Fast unfolding of communities in large networks. J. Stat. Mech. 2008, 2008, P10008. [Google Scholar] [CrossRef]
 Barber, M.J.; Clark, J.W. Detecting network communities by propagating labels under constraints. Phys. Rev. E 2009, 80, 026129. [Google Scholar] [CrossRef]
 Leung, I.X.Y.; Hui, P.; Liò, P.; Crowcroft, J. Towards realtime community detection in large networks. Phys. Rev. E 2009, 79, 066107. [Google Scholar] [CrossRef]
 Barnes, E.R. An algorithm for partitioning the nodes of a graph. In Proceedings of the 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, San Diego, CA, USA, 16–18 December 1981; pp. 303–304. [Google Scholar] [CrossRef]
 Mowshowitz, A.; Dehmer, M. Entropy and the Complexity of Graphs Revisited. Entropy 2012, 14, 559–570. [Google Scholar] [CrossRef][Green Version]
 VazquezAraujo, F.; Dapena, A.; SoutoSalorio, M.J.; Castro, P.M. Calculation of the Connected Dominating Set Considering Vertex Importance Metrics. Entropy 2018, 20, 87. [Google Scholar] [CrossRef]
 Sánchez, P.I.; Müller, E.; Korn, U.L.; Böhm, K.; Kappes, A.; Hartmann, T.; Wagner, D. Efficient Algorithms for a Robust ModularityDriven Clustering of Attributed Graphs. In Proceedings of the 2015 SIAM International Conference on Data Mining, Vancouver, BC, Canada, 30 April–2 May 2015; pp. 100–108. [Google Scholar] [CrossRef]
 Meng, F.; Rui, X.; Wang, Z.; Xing, Y.; Cao, L. Coupled Node Similarity Learning for Community Detection in Attributed Networks. Entropy 2018, 20, 471. [Google Scholar] [CrossRef]
 Fan, X.; Xu, R.Y.D.; Cao, L.; Song, Y. Learning Nonparametric Relational Models by Conjugately Incorporating Node Information in a Network. IEEE Trans. Cybern. 2017, 47, 589–599. [Google Scholar] [CrossRef] [PubMed]
 Xu, Z.; Ke, Y.; Wang, Y.; Cheng, H.; Cheng, J. A modelbased approach to attributed graph clustering. In Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, Scottsdale, AZ, USA, 20–24 May 2012; ACM: New York, NY, USA, 2012; pp. 505–516. [Google Scholar]
 Cruz, J.D.; Bothorel, C.; Poulet, F. Semantic Clustering of Social Networks using Points of View. In Proceedings of the Conference on Information Research and Applications (CORIA 2011), Avignon, France, 16–18 March 2011; pp. 175–182. [Google Scholar]
 Combe, D.; Largeron, C.; EgyedZsigmond, E.; Géry, M. Combining relations and text in scientific network clustering. In Proceedings of the 2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), Istanbul, Turkey, 26–29 August 2012; pp. 1248–1253. [Google Scholar]
 Henderson, T.C.; Simmons, R.; Sacharny, D.; Mitiche, A.; Fan, X. A probabilistic logic for multisource heterogeneous information fusion. In Proceedings of the 2017 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI), Daegu, Korea, 16–18 November 2017; pp. 530–535. [Google Scholar] [CrossRef]
 Wang, Q.; Jiang, J.; Shi, Z.; Wang, W.; Lv, B.; Qi, B.; Yin, Q. A Novel Multisource Fusion Model for Known and Unknown Attack Scenarios. In Proceedings of the 2018 17th IEEE International Conference on Trust, Security and Privacy in Computing and Communications/12th IEEE International Conference on Big Data Science and Engineering (TrustCom/BigDataSE), New York, NY, USA, 1–3 August 2018; pp. 727–736. [Google Scholar]
 Zhang, J.; Philip, S.Y.; Lv, Y. Enterprise community detection. In Proceedings of the 2017 IEEE 33rd International Conference on Data Engineering (ICDE), San Diego, CA, USA, 19–22 April 2017; pp. 219–222. [Google Scholar]
 Neville, J.; Adler, M.; Jensen, D. Clustering relational data using attribute and link information. In Proceedings of the Text Mining and Link Analysis Workshop, 18th International Joint Conference on Artificial Intelligence, Acapulco, Mexico, 9–15 August 2003; pp. 9–15. [Google Scholar]
 Steinhaeuser, K.; Chawla, N.V. Community detection in a large realworld social network. In Social Computing, Behavioral Modeling, and Prediction; Springer: Boston, MA, USA, 2008; pp. 168–175. [Google Scholar]
 Dang, T.; Viennet, E. Community detection based on structural and attribute similarities. In The Sixth International Conference on Digital Society (ICDS); IARIA: Valencia, Spain, 2012; pp. 7–12. [Google Scholar]
 Karypis, G.; Kumar, V. A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. Sci. Comput. 1998, 20, 359–392. [Google Scholar] [CrossRef][Green Version]
 Newman, M.E. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 2006, 103, 8577–8582. [Google Scholar] [CrossRef][Green Version]
 Duch, J.; Arenas, A. Community detection in complex networks using extremal optimization. Phys. Rev. E 2005, 72, 027104. [Google Scholar] [CrossRef]
 Li, W.; Schuurmans, D. Modular Community Detection in Networks. In Proceedings of the TwentySecond International Joint Conference on Artificial Intelligence—Volume Volume Two; AAAI Press: Barcelona, Spain, 2011; pp. 1366–1371. [Google Scholar] [CrossRef]
 Lee, J.; Gross, S.P.; Lee, J. Modularity optimization by conformational space annealing. Phys. Rev. E 2012, 85, 056702. [Google Scholar] [CrossRef]
 Bu, Z.; Zhang, C.; Xia, Z.; Wang, J. A fast parallel modularity optimization algorithm (FPMQA) for community detection in online social network. Knowl. Based Syst. 2013, 50, 246–259. [Google Scholar] [CrossRef]
 Richardson, T.; Mucha, P.J.; Porter, M.A. Spectral tripartitioning of networks. Phys. Rev. E 2009, 80, 036111. [Google Scholar] [CrossRef]
 Newman, M.E.J. Spectral methods for community detection and graph partitioning. Phys. Rev. E 2013, 88, 042822. [Google Scholar] [CrossRef][Green Version]
 SalesPardo, M.; Guimerà, R.; Moreira, A.A.; Amaral, L.A.N. Extracting the hierarchical organization of complex systems. Proc. Natl. Acad. Sci. USA 2007, 104, 15224–15229. [Google Scholar] [CrossRef] [PubMed][Green Version]
 Wakita, K.; Tsurumi, T. Finding Community Structure in Megascale Social Networks: [Extended Abstract]. In Proceedings of the 16th International Conference on World Wide Web (WWW ’07), Banff, AB, Canada, 8–12 May 2007; ACM: New York, NY, USA, 2007; pp. 1275–1276. [Google Scholar] [CrossRef]
 Agarwal, G.; Kempe, D. Modularitymaximizing graph communities via mathematical programming. Eur. Phys. J. B 2008, 66, 409–418. [Google Scholar] [CrossRef][Green Version]
 Zhang, S.; Zhao, H. Normalized modularity optimization method for community identification with degree adjustment. Phys. Rev. E 2013, 88, 052802. [Google Scholar] [CrossRef] [PubMed]
 Khadivi, A.; Ajdari Rad, A.; Hasler, M. Network communitydetection enhancement by proper weighting. Phys. Rev. E 2011, 83, 046104. [Google Scholar] [CrossRef] [PubMed]
 Sun, Y.; Danila, B.; Josić, K.; Bassler, K.E. Improved community structure detection using a modified finetuning strategy. Europhys. Lett. 2009, 86, 28004. [Google Scholar] [CrossRef][Green Version]
 Fortunato, S.; Barthélemy, M. Resolution limit in community detection. Proc. Natl. Acad. Sci. USA 2007, 104, 36–41. [Google Scholar] [CrossRef]
 Good, B.H.; de Montjoye, Y.A.; Clauset, A. Performance of modularity maximization in practical contexts. Phys. Rev. E 2010, 81, 046106. [Google Scholar] [CrossRef]
 Zhang, S.; Zhao, H. Community identification in networks with unbalanced structure. Phys. Rev. E 2012, 85, 066114. [Google Scholar] [CrossRef]
 Newman, M.E.J. Community detection and graph partitioning. Europhys. Lett. 2013, 103, 28003. [Google Scholar] [CrossRef]
 Lin, C.C.; Kang, J.R.; Chen, J.Y. An Integer Programming Approach and Visual Analysis for Detecting Hierarchical Community Structures in Social Networks. Inf. Sci. 2015, 299, 296–311. [Google Scholar] [CrossRef]
 Bai, X.; Yang, P.; Shi, X. An overlapping community detection algorithm based on density peaks. Neurocomputing 2017, 226, 7–15. [Google Scholar] [CrossRef]
 Bothorel, C.; Cruz, J.D.; Magnani, M.; Micenkova, B. Clustering attributed graphs: Models, measures and methods. Netw. Sci. 2015, 3, 408–444. [Google Scholar] [CrossRef]
 Zhou, Y.; Cheng, H.; Yu, J.X. Graph clustering based on structural/attribute similarities. Proc. VLDB Endow. 2009, 2, 718–729. [Google Scholar] [CrossRef][Green Version]
 Ge, R.; Ester, M.; Gao, B.J.; Hu, Z.; Bhattacharya, B.; BenMoshe, B. Joint cluster analysis of attribute data and relationship data: The connected kcenter problem, algorithms and applications. ACM Trans. Knowl. Discov. Data 2008, 2, 7. [Google Scholar] [CrossRef]
 Li, H.; Nie, Z.; Lee, W.C.; Giles, L.; Wen, J.R. Scalable community discovery on textual data with relations. In Proceedings of the 17th ACM Conference on Information and Knowledge Management, Napa Valley, CA, USA, 26–30 October 2008; ACM: New York, NY, USA, 2008; pp. 1203–1212. [Google Scholar]
 Liu, Y.; NiculescuMizil, A.; Gryc, W. Topiclink LDA: Joint models of topic and author community. In Proceedings of the 26th Annual International Conference on Machine Learning, Montreal, QC, Canada, 14–18 June 2009; ACM: New York, NY, USA, 2009; pp. 665–672. [Google Scholar]
 Günnemann, S.; Boden, B.; Färber, I.; Seidl, T. Efficient mining of combined subspace and subgraph clusters in graphs with feature vectors. In PacificAsia Conference on Knowledge Discovery and Data Mining; Springer: Berlin/Heidelberg, Germany, 2013; pp. 261–275. [Google Scholar]
 Gunnemann, S.; Farber, I.; Raubach, S.; Seidl, T. Spectral Subspace Clustering for Graphs with Feature Vectors. In Proceedings of the IEEE 13th International Conference on Data Mining, Dallas, TX, USA, 7–10 December 2013; pp. 231–240. [Google Scholar] [CrossRef]
 Fred, A.L.; Jain, A.K. Combining multiple clusterings using evidence accumulation. IEEE Trans. Pattern Anal. Mach. Intell. 2005, 27, 835–850. [Google Scholar] [CrossRef]
 VegaPons, S.; RuizShulcloper, J. A survey of clustering ensemble algorithms. Intern. J. Pattern Recognit. Artif. Intell. 2011, 25, 337–372. [Google Scholar] [CrossRef]
 Grund, T.U.; Densley, J.A. Ethnic homophily and triad closure: Mapping internal gang structure using exponential random graph models. J. Contemp. Crim. Justice 2015, 31, 354–370. [Google Scholar] [CrossRef]
 Descormiers, K.; Morselli, C. Alliances, conflicts, and contradictions in Montreal’s street gang landscape. Int. Crim. Justice Rev. 2011, 21, 297–314. [Google Scholar] [CrossRef]
 Rand, W.M. Objective Criteria for the Evaluation of Clustering Methods. J. Am. Stat. Assoc. 1971, 66, 846–850. [Google Scholar] [CrossRef][Green Version]
 Hubert, L.; Arabie, P. Comparing partitions. J. Classif. 1985, 2, 193–218. [Google Scholar] [CrossRef]
 Danon, L.; DíazGuilera, A.; Duch, J.; Arenas, A. Comparing community structure identification. J. Stat. Mech. 2005, 2005, P09008. [Google Scholar] [CrossRef]
Methods  RI  ARI  NMI  WQ 

LPACNS  0.5855  0.1717  0.3182  0.1024 
BGLLCNS  0.4889  0.0237  0.0014  0.0165 
KmedCNS  0.8019  0.6038  0.6007  0.1345 
WCMFA  0.9150  0.8299  0.7828  0.1970 
$\Delta $  0.1130  0.2261  0.1822  0.0625 
Methods  RI  ARI  NMI  WQ 

LPACNS  0.3788  0.0015  0.0003  0.0000 
BGLLCNS  0.5265  0.0138  0.1108  0.0032 
KmedCNS  0.3753  0.0353  0.0359  0.0005 
WCMFA  0.5514  0.0712  0.1008  0.0487 
$\Delta $  0.0249  0.0359  −0.0100  0.0455 
Methods  RI  ARI  NMI  WQ 

LPACNS  0.5513  0.2340  0.4312  0.0560 
BGLLCNS  0.6639  0.0110  0.2064  0.0283 
KmedCNS  0.7899  0.5146  0.6372  0.1151 
WCMFA  0.8739  0.6973  0.7785  0.3035 
$\Delta $  0.0840  0.1827  0.1413  0.1884 
Candidate Partitions  The Number of Communities in One Partition $\left\mathit{P}\right$  

2  4  8  16  32  
5  0.39  0.38  0.39  0.46  0.52 
10  0.40  0.43  0.46  0.57  0.65 
15  0.41  0.46  0.51  0.66  0.77 
20  0.44  0.51  0.59  0.78  0.94 
25  0.46  0.55  0.64  0.90  1.15 
30  0.49  0.57  0.71  1.09  1.30 
35  0.54  0.60  0.78  1.24  1.56 
40  0.55  0.65  0.81  1.36  1.74 
45  0.58  0.69  0.89  1.54  1.97 
50  0.64  0.78  1.02  1.66  2.29 
55  0.66  0.78  1.27  1.75  2.37 
60  0.69  0.83  1.28  2.05  2.79 
65  0.71  0.89  1.39  2.37  3.28 
70  0.74  0.93  1.53  2.47  3.30 
75  0.78  1.00  1.75  2.52  3.76 
80  0.80  1.02  1.76  2.95  4.13 
85  0.87  1.13  1.94  3.28  4.39 
90  0.89  1.14  2.00  3.49  4.73 
95  0.92  1.19  2.06  3.73  5.17 
100  0.94  1.24  2.37  4.05  5.43 
Node Size  The Number of Communities in One Partition $\left\mathit{P}\right$  

2  4  8  16  32  
200  0.93  1.25  2.30  4.90  5.65 
400  1.37  1.70  2.55  6.32  8.87 
600  1.92  2.14  2.99  6.82  15.50 
800  2.62  2.93  3.75  7.10  23.04 
1000  3.50  3.67  4.62  8.86  35.34 
1200  4.43  4.70  5.59  11.16  37.86 
1400  5.51  5.74  6.49  14.37  49.48 
1600  6.86  7.16  7.72  15.74  59.71 
1800  8.08  8.35  9.00  15.76  82.09 
2000  9.17  9.78  10.86  17.92  92.41 
2200  11.47  11.66  12.18  19.80  118.44 
2400  13.33  13.05  14.09  20.96  126.04 
2600  15.26  15.42  16.01  22.60  145.14 
2800  15.38  17.52  18.34  24.36  149.09 
3000  19.68  19.84  20.66  25.90  162.64 
3200  22.17  22.25  22.56  28.27  174.81 
3400  24.45  24.42  25.76  29.11  179.99 
3600  26.99  27.16  27.24  31.44  184.27 
3800  29.83  29.14  30.22  33.93  187.42 
4000  32.14  31.99  32.98  37.06  243.40 
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Luo, S.; Zhang, Z.; Zhang, Y.; Ma, S. CoAssociation MatrixBased MultiLayer Fusion for Community Detection in Attributed Networks. Entropy 2019, 21, 95. https://doi.org/10.3390/e21010095
Luo S, Zhang Z, Zhang Y, Ma S. CoAssociation MatrixBased MultiLayer Fusion for Community Detection in Attributed Networks. Entropy. 2019; 21(1):95. https://doi.org/10.3390/e21010095
Chicago/Turabian StyleLuo, Sheng, Zhifei Zhang, Yuanjian Zhang, and Shuwen Ma. 2019. "CoAssociation MatrixBased MultiLayer Fusion for Community Detection in Attributed Networks" Entropy 21, no. 1: 95. https://doi.org/10.3390/e21010095