Abstract
In the recent research of data mining, frequent structures in a sequence of graphs have been studied intensively, and one of the main concern is changing structures along a sequence of graphs that can capture dynamic properties of data. On the contrary, we newly focus on “preserving structures” in a graph sequence that satisfy a given property for a certain period, and mining such structures is studied. We bring up two structures of practical importance, a connected vertex subset and a clique that exist for a certain period. We consider the problem of enumerating these structures and present polynomial delay algorithms for the problems. Their running time may depend on the size of the representation, however, if each edge has at most one time interval in the representation, the running time is \(O(|V| |E|^3)\) for connected vertex subsets and \(O(\min \{\Delta ^5, |E|^2 \Delta \})\) for cliques, where the input graph is \(G=(V, E)\) with maximum degree \(\Delta \). To the best of our knowledge, this is the first systematic approach to the treatment of this notion, namely, preserving structures.
A part of this research is supported by JST CREST and Grant-in-Aid for Scientific Research (KAKENHI), No. 23500022 and 15H00853.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arimura, H., Uno, T., Shimozono, S.: Time and space efficient discovery of maximal geometric graphs. In: Corruble, V., Takeda, M., Suzuki, E. (eds.) DS 2007. LNCS (LNAI), vol. 4755, pp. 42–55. Springer, Heidelberg (2007)
Avis, D., Fukuda, K.: Reverse search for enumeration. Discr. Appl. Math. 65, 21–46 (1996)
Berlingerio, M., Bonchi, F., Bringmann, B., Gionis, A.: Mining graph evolution rules. In: Buntine, W., Grobelnik, M., Mladenić, D., Shawe-Taylor, J. (eds.) ECML PKDD 2009, Part I. LNCS, vol. 5781, pp. 115–130. Springer, Heidelberg (2009)
Buchin, K., Buchin, M., van Kreveld, M., Speckmann, B., Staals, F.: Trajectory grouping structure. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 219–230. Springer, Heidelberg (2013)
Xuan, B.B., Ferreira, A., Jarry, A.: Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. of Foundations of Computer Science 14, 267–285 (2003)
Borgwardt, K.M., Kriegel, H.P., Wackersreuther, P.: Pattern mining in frequent dynamic subgraphs. In: Proc. 6th IEEE ICDM, pp. 818–822 (2006)
Eppstein, D., Galil, Z., Italiano, G.F., Nissenzweig, A.: Sparsification–A technique for speeding up dynamic graph algorithms. J. ACM 44, 669–696 (1997)
Fukuda, K., Matsui, T.: Finding all the perfect matchings in bipartite graphs. Applied Mathematics Letters 7, 15–18 (1994)
Han, J., Dong, G., Yin, Y.: Efficient mining of partial periodic patterns in time series database. In: Proc. 15th IEEE ICDE, pp. 106–115 (1999)
Inokuchi A., Washio, T.: A fast method to mine frequent subsequences from graph sequence data. In: Proc. 8th IEEE ICDM, pp. 303–312 (2008)
Mining graph data: A. Inokuchi T. Washio and H. Motoda. Complete mining of frequent patterns from graphs. Machine Learning 50, 321–354 (2003)
Kalnis, P., Mamoulis, N., Bakiras, S.: On discovering moving clusters in spatio-temporal data. In: Medeiros, C.B., Egenhofer, M., Bertino, E. (eds.) SSTD 2005. LNCS, vol. 3633, pp. 364–381. Springer, Heidelberg (2005)
Lahiri, M. Berger-Wolf, T.Y.: Mining periodic behavior in dynamic social networks. In: Proc. 8th IEEE ICDM, pp. 373–382 (2008)
Li, Z., Ding, B., Han, J., Kays, R.: Swarm: Mining relaxed temporal moving object clusters. In: Proc. 36th Int’l Conf. on VLDB, pp. 723–734 (2010)
Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Heidelberg (2004)
Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Efficient mining of association rules using closed itemset lattices. J. Information Systems 24, 25–46 (1999)
Read, R.C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5, 237–252 (1975)
Sun, J., Faloutsos, C., Papadimitriou, S., Yu, P.S.: GraphScope: Parameter-free mining of large time-evolving graphs. In: Proc. 13th ACM Int’l Conf. on KDD, pp. 687–696 (2007)
Tantipathananandh, C., Berger-Wolf, T.: Constant-factor approximation algorithms for identifying dynamic communities. In: Proc. 15th ACM Int’l Conf. on KDD, pp. 827–836 (2009)
Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments. Theor. Comp. Sci. 363, 28–42 (2006)
Uno, Y., Ota, Y., Uemichi, A.: Web structure mining by isolated cliques. IEICE Transactions on Information and Systems E90–D, 1998–2006 (2007)
Yan, X., Han, J.: GSPAN: graph-based substructure pattern mining. In: Proc. 2nd IEEE ICDM, pp. 721–724 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Uno, T., Uno, Y. (2015). Mining Preserving Structures in a Graph Sequence. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-21398-9_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21397-2
Online ISBN: 978-3-319-21398-9
eBook Packages: Computer ScienceComputer Science (R0)