Abstract
Dense subtensor detection gains remarkable success in spotting anomaly and fraudulent behaviors for the multi-aspect data (i.e., tensors), like in social media and event streams. Existing methods detect the densest subtensors flatly and separately, with an underlying assumption that these subtensors are exclusive. However, many real-world scenario usually present hierarchical properties, e.g., the core-periphery structure or dynamic communities in networks. In this paper, we propose CatchCore, a novel framework to effectively find the hierarchical dense subtensors. We first design a unified metric for dense subtensor detection, which can be optimized with gradient-based methods. With the proposed metric, CatchCore detects hierarchical dense subtensors through the hierarchy-wise alternative optimization. Finally, we utilize the minimum description length principle to measure the quality of detection result and select the optimal hierarchical dense subtensors. Extensive experiments on synthetic and real-world datasets demonstrate that CatchCore outperforms the top competitors in accuracy for detecting dense subtensors and anomaly patterns. Additionally, CatchCore identified a hierarchical researcher co-authorship group with intense interactions in DBLP dataset. Also CatchCore scales linearly with all aspects of tensors.
Code of this paper is available at: http://github.com/wenchieh/catchcore.
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Notes
- 1.
Entrywise, the n-mode product between the tensor \(\varvec{\mathscr {R}}\) and vector \(\varvec{x}\) can be denoted as:
. - 2.
We use \(\, \bar{\times }_{(-n)} \,\) to denote conducting full-mode product except the n-th mode.
- 3.
More generally, we can also set different density ratios between hierarchies rather than the fixed one parameter for specific concern.
- 4.
\(\log ^{*} x\) is the universal code length for an integer x [18].
.References
Akoglu, L., Tong, H., Koutra, D.: Graph based anomaly detection and description: a survey. In: Data Mining and Knowledge Discovery (2015)
Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. WAW
Balalau, O.D., Bonchi, F., Chan, T.H.H., Gullo, F., Sozio, M.: Finding subgraphs with maximum total density and limited overlap. In: WSDM 2015 (2015)
Chen, J., Saad, Y.: Dense subgraph extraction with application to community detection. IEEE Trans. Knowl. Eng. 24(7), 1216–1230 (2010)
Coleman, T.F., Li, Y.: An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6(2), 418–445 (1996)
Edler, D., Bohlin, L., et al.: Mapping higher-order network flows in memory and multilayer networks with infomap. Algorithms 10(4), 112 (2017)
Gibson, D., Kumar, R., Tomkins, A.: Discovering large dense subgraphs in massive graphs. In: VLDB 2005. VLDB Endowment (2005)
Gorovits, A., Gujral, E., Papalexakis, E.E., Bogdanov, P.: LARC: learning activity-regularized overlapping communities across time. In: SIGKDD 2018. ACM (2018)
Grippo, L., Sciandrone, M.: On the convergence of the block nonlinear Gauss-Seidel method under convex constraints. Oper. Res. Lett. 26(3), 127–136 (2000)
Hooi, B., Song, H.A., Beutel, A., Shah, N., Shin, K., Faloutsos, C.: FRAUDAR: bounding graph fraud in the face of camouflage. In: SIGKDD 2016, pp. 895–904 (2016)
Jiang, M., Beutel, A., Cui, P., Hooi, B., Yang, S., Faloutsos, C.: A general suspiciousness metric for dense blocks in multimodal data. In: ICDM 2015 (2015)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. In: SIAM (2009)
Kumar, R., Novak, J., Tomkins, A.: Structure and evolution of online social networks. In: Link Mining: Models, Algorithms, and Applications (2010)
Leskovec, J., Lang, K.J., Dasgupta, A., Mahoney, M.W.: Statistical properties of community structure in large social and information networks. In: WWW (2008)
Lin, C.J.: Projected gradient methods for nonnegative matrix factorization. Neural Comput. 19(10), 2756–2779 (2007)
Lin, C.J., Moré, J.J.: Newton’s method for large bound-constrained optimization problems. SIAM J. Optim. 9(4), 1100–1127 (1999)
Papadimitriou, S., Sun, J., Faloutsos, C., Yu, P.S.: Hierarchical, parameter-free community discovery. In: Daelemans, W., Goethals, B., Morik, K. (eds.) ECML PKDD 2008. LNCS (LNAI), vol. 5212, pp. 170–187. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-87481-2_12
Rissanen, J.: A universal prior for integers and estimation by minimum description length. Ann. Stat. 11(2), 416–431 (1983)
Sariyüce, A.E., Pinar, A.: Fast hierarchy construction for dense subgraphs. VLDB
Shin, K., Hooi, B., Faloutsos, C.: M-zoom: fast dense-block detection in tensors with quality guarantees. In: Frasconi, P., Landwehr, N., Manco, G., Vreeken, J. (eds.) ECML PKDD 2016. LNCS (LNAI), vol. 9851, pp. 264–280. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-46128-1_17
Shin, K., Hooi, B., Kim, J., Faloutsos, C.: D-cube: Dense-block detection in terabyte-scale tensors. In: WSDM 2017. ACM (2017)
Shin, K., Hooi, B., Kim, J., Faloutsos, C.: DenseAlert: incremental dense-subtensor detection in tensor streams (2017)
Siddique, B., Akhtar, N.: Temporal hierarchical event detection of timestamped data. In: ICCCA 2017 (2017)
Tsourakakis, C., Bonchi, F., Gionis, A., Gullo, F., Tsiarli, M.: Denser than the densest subgraph: extracting optimal quasi-cliques with quality guarantees. In: SIGKDD 2013. ACM (2013)
Yang, B., Di, J., Liu, J., Liu, D.: Hierarchical community detection with applications to real-world network analysis. In: DKE (2013)
Zhang, S., et al.: Hidden: hierarchical dense subgraph detection with application to financial fraud detection. In: SDM 2017. SIAM (2017)
Acknowledgments
This material is based upon work supported by the Strategic Priority Research Program of CAS (XDA19020400), NSF of China (61772498, 61425016, 61872206), and the Beijing NSF (4172059).
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Feng, W., Liu, S., Cheng, X. (2020). CatchCore: Catching Hierarchical Dense Subtensor. In: Brefeld, U., Fromont, E., Hotho, A., Knobbe, A., Maathuis, M., Robardet, C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2019. Lecture Notes in Computer Science(), vol 11906. Springer, Cham. https://doi.org/10.1007/978-3-030-46150-8_10
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