Abstract
Social networks are becoming larger and more complex as new ways of collecting social interaction data arise (namely from online social networks, mobile devices sensors, ...). These networks are often large-scale and of high dimensionality. Therefore, dealing with such networks became a challenging task. An intuitive way to deal with this complexity is to resort to tensors. In this context, the application of tensor decomposition has proven its usefulness in modelling and mining these networks: it has not only been applied for exploratory analysis (thus allowing the discovery of interaction patterns), but also for more demanding and elaborated tasks such as community detection and link prediction. In this work, we provide an overview of the methods based on tensor decomposition for the purpose of analysing time-evolving social networks from various perspectives: from community detection, link prediction and anomaly/event detection to network summarization and visualization. In more detail, we discuss the ideas exploited to carry out each social network analysis task as well as its limitations in order to give a complete coverage of the topic.











Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Acar E, Kolda TG, Dunlavy DM (2011) All-at-once optimization for coupled matrix and tensor factorizations. arXiv preprint arXiv:11053422
Ahn KJ, Guha S, McGregor A (2012) Graph sketches: sparsification, spanners, and subgraphs. In: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI symposium on principles of database systems, pp 5–14
Akoglu L, Tong H, Koutra D (2015) Graph based anomaly detection and description: a survey. Data Min Knowl Discov 29(3):626–688
Al-Sharoa E, Al-khassaweneh M, Aviyente S (2017) A tensor based framework for community detection in dynamic networks. In: 2017 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, pp 2312–2316
Andersson CA, Bro R (2000) The n-way toolbox for matlab. Chemom Intell Lab Syst 52(1):1–4
Araujo MR, Ribeiro PMP, Faloutsos C (2017) Tensorcast: forecasting with context using coupled tensors (best paper award). In: 2017 IEEE international conference on data mining (ICDM). IEEE, pp 71–80
Araujo M, Papadimitriou S, Günnemann S, Faloutsos C, Basu P, Swami A, Papalexakis EE, Koutra D (2014) Com2: fast automatic discovery of temporal (‘comet’) communities. In: Pacific-Asia conference on knowledge discovery and data mining. Springer, pp 271–283
Austin W, Ballard G, Kolda TG (2016) Parallel tensor compression for large-scale scientific data. In: 2016 IEEE international parallel and distributed processing symposium. IEEE, pp 912–922
Bader BW, Harshman RA, Kolda TG (2007) Temporal analysis of semantic graphs using asalsan. In: ICDM. IEEE, pp 33–42
Bader BW, Kolda TG (2008) Efficient matlab computations with sparse and factored tensors. SIAM J Sci Comput 30(1):205–231
Bak P, Paczuski M, Shubik M (1996) Price variations in a stock market with many agents. arXiv preprint arXiv:cond-mat/9609144
Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512
Bauer F, Lizier JT (2012) Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: a walk counting approach. EPL (Europhys Lett) 99(6):68007
Beutel A, Talukdar PP, Kumar A, Faloutsos C, Papalexakis EE, Xing EP (2014) Flexifact: scalable flexible factorization of coupled tensors on hadoop. In: Proceedings of the 2014 SIAM international conference on data mining. SIAM, pp 109–117
Billio M, Getmansky M, Lo AW, Pelizzon L (2012) Econometric measures of connectedness and systemic risk in the finance and insurance sectors. J Financ Econ 104(3):535–559
Boldi P, Vigna S (2004) The webgraph framework I: compression techniques. In: Proceedings of the 13th international conference on World Wide Web. ACM, pp 595–602
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J et al (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3(1):1–122
Brett W, Bader TGK et al (2020) Matlab tensor toolbox, version. https://www.tensortoolbox.org
Bro R, De Jong S (1997) A fast non-negativity-constrained least squares algorithm. J Chemom J Chemom Soc 11(5):393–401
Bro R, Kiers HA (2003) A new efficient method for determining the number of components in parafac models. J Chemom J Chemom Soc 17(5):274–286
Brunetti C, Harris JH, Mankad S, Michailidis G (2019) Interconnectedness in the interbank market. J Financ Econ 133(2):520–538
Carroll JD, Chang JJ (1970) Analysis of individual differences in multidimensional scaling via an n-way generalization of “eckart-young” decomposition. Psychometrika 35(3):283–319
Ceulemans E, Kiers HA (2006) Selecting among three-mode principal component models of different types and complexities: a numerical convex hull based method. Br J Math Stat Psychol 59(1):133–150
Chen H, Chung W, Qin J, Reid E, Sageman M, Weimann G (2008) Uncovering the dark web: a case study of jihad on the web. J Am Soc Inf Sci Technol 59(8):1347–1359
Chi EC, Kolda TG (2012) On tensors, sparsity, and nonnegative factorizations. SIAM J Matrix Anal Appl 33(4):1272–1299
Choi JH, Vishwanathan S (2014) Dfacto: distributed factorization of tensors. In: Advances in neural information processing systems, pp 1296–1304
da Silva Fernandes S, Tork HF, da Gama JMP (2017) The initialization and parameter setting problem in tensor decomposition-based link prediction. In: 2017 IEEE international conference on data science and advanced analytics (DSAA), pp 99–108
Drineas P, Mahoney MW (2007) A randomized algorithm for a tensor-based generalization of the singular value decomposition. Linear Algebra Appl 420(2–3):553–571
Dunlavy DM, Kolda TG, Acar E (2011) Temporal link prediction using matrix and tensor factorizations. ACM Trans Knowl Discov Data (TKDD) 5(2):10
Erdos D, Miettinen P (2013) Discovering facts with Boolean tensor tucker decomposition. In: Proceedings of the 22nd ACM international conference on conference on information and knowledge management. ACM, pp 1569–1572
Faloutsos M, Faloutsos P, Faloutsos C (1999) On power-law relationships of the internet topology. SIGCOMM Comput Commun Rev 29(4):251–262
Fanaee-T H, Gama J (2016a) Event detection from traffic tensors: a hybrid model. Neurocomputing 203:22–33
Fanaee-T H, Gama J (2016b) Tensor-based anomaly detection: an interdisciplinary survey. Knowl Based Syst 98:130–147
Fernandes S, Fanaee-T H, Gama J (2018) Dynamic graph summarization: a tensor decomposition approach. Data Min Knowl Discov 32(5):1397–1420
Fernandes S, Fanaee-T H, Gama J (2019) Evolving social networks analysis via tensor decompositions: from global event detection towards local pattern discovery and specification. In: International conference on discovery science. Springer, pp 385–395
Ferrara E, De Meo P, Catanese S, Fiumara G (2014) Detecting criminal organizations in mobile phone networks. Expert Syst Appl 41(13):5733–5750
Fortunato S (2010) Community detection in graphs. Phys Rep 486(3–5):75–174
Gahrooei MR, Paynabar K (2018) Change detection in a dynamic stream of attributed networks. J Qual Technol 50(4):418–430
Gauvin L, Panisson A, Cattuto C (2014) Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach. PloS ONE 9(1):e86028
Génois M, Vestergaard CL, Fournet J, Panisson A, Bonmarin I, Barrat A (2015) Data on face-to-face contacts in an office building suggest a low-cost vaccination strategy based on community linkers. Netw Sci 3:326–347
Goldfarb D, Qin Z (2014) Robust low-rank tensor recovery: models and algorithms. SIAM J Matrix Anal Appl 35(1):225–253
Goñi J, Esteban FJ, de Mendizábal NV, Sepulcre J, Ardanza-Trevijano S, Agirrezabal I, Villoslada P (2008) A computational analysis of protein-protein interaction networks in neurodegenerative diseases. BMC Syst Biol 2(1):52
Görlitz O, Sizov S, Staab S (2008) Pints: peer-to-peer infrastructure for tagging systems. In: IPTPS, p 19
Gorovits A, Gujral E, Papalexakis EE, Bogdanov P (2018) Larc: learning activity-regularized overlapping communities across time. In: Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 1465–1474
Grünwald PD (2007) The minimum description length principle. MIT Press, Cambridge
Harshman RA (1970) Foundations of the PARAFAC procedure: models and conditions for an “explanatory” multi-modal factor analysis. In: UCLA working papers in phonetics, vol 16, no 1, p 84
Heiberger RH (2018) Predicting economic growth with stock networks. Phys Stat Mech Appl 489:102–111
Hitchcock FL (1927) The expression of a tensor or a polyadic as a sum of products. J Math Phys 6(1–4):164–189
Isella L, Stehlé J, Barrat A, Cattuto C, Pinton JF, Van den Broeck W (2011) What’s in a crowd? Analysis of face-to-face behavioral networks. J Theor Biol 271(1):166–180
Jeon I, Papalexakis EE, Kang U, Faloutsos C (2015) Haten2: billion-scale tensor decompositions. In: 2015 IEEE 31st international conference on data engineering (ICDE). IEEE, pp 1047–1058
Jeon B, Jeon I, Sael L, Kang U (2016) Scout: scalable coupled matrix-tensor factorization-algorithm and discoveries. In: IEEE 32nd international conference on data engineering (ICDE), 2016. IEEE, pp 811–822
Jordán F, Nguyen TP, Wc L (2012) Studying protein–protein interaction networks: a systems view on diseases. Brief Funct Genom 11(6):497–504
Kang U, Papalexakis E, Harpale A, Faloutsos C (2012) Gigatensor: scaling tensor analysis up by 100 times-algorithms and discoveries. In: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, pp 316–324
Keeling MJ, Eames KT (2005) Networks and epidemic models. J R Soc Interface 2(4):295–307
Kiers HA (1993) An alternating least squares algorithm for parafac2 and three-way dedicom. Comput Stat Data Anal 16(1):103–118
Kiers HAL (2000) Towards a standardized notation and terminology in multiway analysis. J Chemom 14(3):105–122
Kiers HA, Kinderen A (2003) A fast method for choosing the numbers of components in tucker3 analysis. Br J Math Stat Psychol 56(1):119–125
Kolda TG, Sun J (2008) Scalable tensor decompositions for multi-aspect data mining. In: Eighth IEEE international conference on data mining, 2008. ICDM’08. IEEE, pp 363–372
Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev 51(3):455–500
Kossaifi J, Panagakis Y, Anandkumar A, Pantic M (2019) Tensorly: tensor learning in python. J Mach Learn Res 20(1):925–930
Koutra D, Papalexakis EE, Faloutsos C (2012) Tensorsplat: spotting latent anomalies in time. In: 2012 16th Panhellenic conference on informatics (PCI). IEEE, pp 144–149
Kwak H, Lee C, Park H, Moon S (2010) What is twitter, a social network or a news media? In: Proceedings of the 19th international conference on World wide web. ACM, pp 591–600
LeFevre K, Terzi E (2010) Grass: graph structure summarization. In: Proceedings of the 2010 SIAM international conference on data mining. SIAM, pp 454–465
Leskovec J, Adamic LA, Huberman BA (2007) The dynamics of viral marketing. ACM Trans Web (TWEB) 1(1):5
Leskovec J, Kleinberg J, Faloutsos C (2005) Graphs over time: densification laws, shrinking diameters and possible explanations. In: Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining. ACM, pp 177–187
Leskovec J, Mcauley JJ (2012) Learning to discover social circles in ego networks. In: Advances in neural information processing systems, pp 539–547
Ley M (2002) The dblp computer science bibliography: evolution, research issues, perspectives. In: International symposium on string processing and information retrieval. Springer, pp 1–10
Li J, Bien J, Wells M, Li MJ (2018) Package ‘rtensor’. J Stat Softw 87:1–31
Lin YR, Sun J, Castro P, Konuru R, Sundaram H, Kelliher A (2009) Metafac: community discovery via relational hypergraph factorization. In: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, pp 527–536
Liu J, Musialski P, Wonka P, Ye J (2013) Tensor completion for estimating missing values in visual data. IEEE Trans Pattern Anal Mach Intell 35(1):208–220
Liu K, Da Costa JPCL, So HC, Huang L, Ye J (2016) Detection of number of components in candecomp/parafac models via minimum description length. Digit Signal Process Rev J 51:110–123
Liu Y, Safavi T, Dighe A, Koutra D (2018) Graph summarization methods and applications: a survey. ACM Comput Surv (CSUR) 51(3):62
Lü L, Zhou T (2011) Link prediction in complex networks: a survey. Phys Stat Mech Appl 390(6):1150–1170
Ma X, Dong D (2017) Evolutionary nonnegative matrix factorization algorithms for community detection in dynamic networks. IEEE Trans Knowl Data Eng 29(5):1045–1058
Mankad S, Michailidis G (2013) Structural and functional discovery in dynamic networks with non-negative matrix factorization. Phys Rev E 88(4):042812
Maruhashi K, Guo F, Faloutsos C (2011) Multiaspectforensics: pattern mining on large-scale heterogeneous networks with tensor analysis. In: 2011 International conference on advances in social networks analysis and mining (ASONAM). IEEE, pp 203–210
Mastrandrea R, Fournet J, Barrat A (2015) Contact patterns in a high school: a comparison between data collected using wearable sensors, contact diaries and friendship surveys. PloS ONE 10(9):e0136497
Matsubara Y, Sakurai Y, Faloutsos C, Iwata T, Yoshikawa M (2012) Fast mining and forecasting of complex time-stamped events. In: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, pp 271–279
Michalski R, Palus S, Kazienko P (2011) Matching organizational structure and social network extracted from email communication. In: Lecture notes in business information processing, vol 87. Springer, Berlin, pp 197–206
Milgram S (1967) The small world problem. Psychol Today 2:60–67
Mørup M, Hansen LK (2009) Automatic relevance determination for multi-way models. J Chemom 23(7–8):352–363
Mørup M, Hansen LK, Arnfred SM (2008) Algorithms for sparse nonnegative tucker decompositions. Neural Comput 20(8):2112–2131
Oliveira M, Gama J (2011) Visualizing the evolution of social networks. In: Portuguese conference on artificial intelligence. Springer, pp 476–490
Papalexakis EE (2016) Automatic unsupervised tensor mining with quality assessment. In: Proceedings of the 2016 SIAM international conference on data mining. SIAM, pp 711–719
Papalexakis EE, Faloutsos C (2015) Fast efficient and scalable core consistency diagnostic for the parafac decomposition for big sparse tensors. In: ICASSP, pp 5441–5445
Papalexakis EE, Faloutsos C, Sidiropoulos ND (2012) Parcube: sparse parallelizable tensor decompositions. In: Joint European conference on machine learning and knowledge discovery in databases. Springer, pp 521–536
Papalexakis EE, Sidiropoulos ND, Bro R (2013) From k-means to higher-way co-clustering: multilinear decomposition with sparse latent factors. IEEE Trans Signal Process 61(2):493–506
Papalexakis EE, Faloutsos C, Sidiropoulos ND (2016) Tensors for data mining and data fusion: models, applications, and scalable algorithms. ACM Trans Intell Syst Technol 8(2):1–44
Papalexakis E, Pelechrinis K, Faloutsos C (2014) Spotting misbehaviors in location-based social networks using tensors. In: Proceedings of the 23rd international conference on world wide web. ACM, pp 551–552
Park N, Jeon B, Lee J, Kang U (2016) Bigtensor: mining billion-scale tensor made easy. In: Proceedings of the 25th ACM international on conference on information and knowledge management. ACM, pp 2457–2460
Pasricha R, Gujral E, Papalexakis EE (2018) Identifying and alleviating concept drift in streaming tensor decomposition. arXiv preprint arXiv:180409619
Pavlopoulos GA, Secrier M, Moschopoulos CN, Soldatos TG, Kossida S, Aerts J, Schneider R, Bagos PG (2011) Using graph theory to analyze biological networks. BioData Min 4(1):1–27
Peng W, Li T (2011) Temporal relation co-clustering on directional social network and author-topic evolution. Knowl Inf Syst 26(3):467–486
Phan AH, Cichocki A (2011) Parafac algorithms for large-scale problems. Neurocomputing 74(11):1970–1984
Priebe CE, Conroy JM, Marchette DJ, Park Y (2005) Scan statistics on enron graphs. Comput Math Organ Theory 11(3):229–247
Ranshous S, Shen S, Koutra D, Harenberg S, Faloutsos C, Samatova NF (2015) Anomaly detection in dynamic networks: a survey. Wiley Interdiscip Rev Comput Stat 7(3):223–247
Rayana S, Akoglu L (2014) An ensemble approach for event detection and characterization in dynamic graphs. In: Proceedings of the 2nd ACM SIGKDD workshop on outlier detection and description under data diversity (ODD)
Rossetti G, Cazabet R (2018) Community discovery in dynamic networks: a survey. ACM Comput Surv (CSUR) 51(2):35
Sapienza A, Panisson A, Wu J, Gauvin L, Cattuto C (2015) Anomaly detection in temporal graph data: an iterative tensor decomposition and masking approach. In: International workshop on advanced analytics and learning on temporal data AALTD, Porto. ECML PKDD, Portugal
Shah N, Koutra D, Zou T, Gallagher B, Faloutsos C (2015) Timecrunch: interpretable dynamic graph summarization. In: Proceedings of the 21th ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 1055–1064
Shashua A, Hazan T (2005) Non-negative tensor factorization with applications to statistics and computer vision. In: Proceedings of the 22nd international conference on Machine learning. ACM, pp 792–799
Sheikholeslami F, Giannakis GB (2017) Identification of overlapping communities via constrained egonet tensor decomposition. arXiv preprint arXiv:170704607
Shi L, Gangopadhyay A, Janeja VP (2015) Stensr: spatio-temporal tensor streams for anomaly detection and pattern discovery. Knowl Inf Syst 43(2):333–353
Sidiropoulos ND, De Lathauwer L, Fu X, Huang K, Papalexakis EE, Faloutsos C (2017) Tensor decomposition for signal processing and machine learning. IEEE Trans Signal Process 65(13):3551–3582
Spiegel S, Clausen J, Albayrak S, Kunegis J (2011) Link prediction on evolving data using tensor factorization. In: New frontiers in applied data mining. Springer, pp 100–110
Sun J, Tao D, Papadimitriou S, Yu PS, Faloutsos C (2008) Incremental tensor analysis: theory and applications. ACM Trans Knowl Discov Data 2(3):11:1–11:37
Sun J, Papadimitriou S, Lin CY, Cao N, Liu S, Qian W (2009) Multivis: content-based social network exploration through multi-way visual analysis. In: Proceedings of the 2009 SIAM international conference on data mining. SIAM, pp 1064–1075
Sun J, Tao D, Faloutsos C (2006) Beyond streams and graphs: dynamic tensor analysis. In: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, pp 374–383
Tabassum S, Pereira FSF, Fernandes S, Gama J (2018) Social network analysis: an overview. Wiley Interdiscip Rev Data Min Knowl Discov 8(5):e1256
Tan H, Feng G, Feng J, Wang W, Zhang YJ, Li F (2013a) A tensor-based method for missing traffic data completion. Transp Res C Emerg Technol 28:15–27
Tan H, Feng J, Feng G, Wang W, Zhang YJ (2013b) Traffic volume data outlier recovery via tensor model. Math Probl Eng 2013. https://doi.org/10.1155/2013/164810
Tang J, Sun J, Wang C, Yang Z (2009) Social influence analysis in large-scale networks. In: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, pp 807–816
Timmerman ME, Kiers HA (2000) Three-mode principal components analysis: choosing the numbers of components and sensitivity to local optima. Br J Math Stat Psychol 53(1):1–16
Tsalouchidou I, Bonchi F, Morales GDF, Baeza-Yates R (2018) Scalable dynamic graph summarization. IEEE Trans Knowl Data Eng 32(2):360–373
Tucker LR (1966) Some mathematical notes on three-mode factor analysis. Psychometrika 31(3):279–311
Vanhems P, Barrat A, Cattuto C, Pinton JF, Khanafer N, Régis C, Ba K, Comte B, Voirin N (2013) Estimating potential infection transmission routes in hospital wards using wearable proximity sensors. PloS ONE 8(9):e73970
Vervliet N, Debals O, Sorber L, Van Barel M, De Lathauwer L (2016) Tensorlab 3.0. https://www.tensorlab.net
Viswanath B, Mislove A, Cha M, Gummadi KP (2009) On the evolution of user interaction in facebook. In: Proceedings of the 2nd ACM SIGCOMM workshop on social networks (WOSN’09)
Von Luxburg U (2007) A tutorial on spectral clustering. Stat Comput 17(4):395–416
Wang P, Xu B, Wu Y, Zhou X (2015) Link prediction in social networks: the state-of-the-art. Sci China Inf Sci 58(1):1–38
Xie K, Li X, Wang X, Xie G, Wen J, Cao J, Zhang D (2017) Fast tensor factorization for accurate internet anomaly detection. IEEE/ACM Trans Netw 25(6):3794–3807
Xu J, Chen H (2005) Criminal network analysis and visualization. Commun ACM 48(6):100–107
Zou B, Li C, Tan L, Chen H (2015) Gputensor: efficient tensor factorization for context-aware recommendations. Inf Sci 299:159–177
Acknowledgements
This work is financed by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia within project UIDB/50014/2020. Sofia Fernandes also acknowledges the support of FCT via the PhD scholarship PD/BD/114189/2016.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fernandes, S., Fanaee-T, H. & Gama, J. Tensor decomposition for analysing time-evolving social networks: an overview. Artif Intell Rev 54, 2891–2916 (2021). https://doi.org/10.1007/s10462-020-09916-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10462-020-09916-4