Comparing temporal graphs using dynamic time warping

Within many real-world networks, the links between pairs of nodes change over time. Thus, there has been a recent boom in studying temporal graphs. Recognizing patterns in temporal graphs requires a proximity measure to compare different temporal graphs. To this end, we propose to study dynamic time warping on temporal graphs. We define the dynamic temporal graph warping (dtgw) distance to determine the dissimilarity of two temporal graphs. Our novel measure is flexible and can be applied in various application domains. We show that computing the dtgw-distance is a challenging (in general) NP -hard optimization problem and identify some polynomial-time solvable special cases. Moreover, we develop a quadratic programming formulation and an efficient heuristic. In experiments on real-world data, we show that the heuristic performs very well and that our dtgw-distance performs favorably in de-anonymizing networks compared to other approaches.

[1]  Eamonn J. Keogh,et al.  Searching and Mining Trillions of Time Series Subsequences under Dynamic Time Warping , 2012, KDD.

[2]  A. Volgenant,et al.  A shortest augmenting path algorithm for dense and sparse linear assignment problems , 1987, Computing.

[3]  Eamonn J. Keogh,et al.  Experimental comparison of representation methods and distance measures for time series data , 2010, Data Mining and Knowledge Discovery.

[4]  Mahantapas Kundu,et al.  The journey of graph kernels through two decades , 2018, Comput. Sci. Rev..

[5]  Rolf Niedermeier,et al.  Computing Maximum Matchings in Temporal Graphs , 2019, STACS.

[6]  Brijnesh J. Jain,et al.  On the geometry of graph spaces , 2016, Discret. Appl. Math..

[7]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[8]  Micha Sharir,et al.  Dynamic Time Warping and Geometric Edit Distance: Breaking the Quadratic Barrier , 2016, ICALP.

[9]  Ryan A. Rossi,et al.  Continuous-Time Dynamic Network Embeddings , 2018, WWW.

[10]  Gustavo E. A. P. A. Batista,et al.  Speeding Up All-Pairwise Dynamic Time Warping Matrix Calculation , 2016, SDM.

[11]  Binh-Minh Bui-Xuan,et al.  Temporal Matching , 2018, Theor. Comput. Sci..

[12]  Salvatore Tabbone,et al.  Graph Matching Based on Node Signatures , 2009, GbRPR.

[13]  Christina Boucher,et al.  Co-evolving Patterns in Temporal Networks of Varying Evolution , 2019, BCB.

[14]  Vassilis Kostakos.  Temporal Graphs , 2014, Encyclopedia of Social Network Analysis and Mining.

[15]  Nils M. Kriege,et al.  A survey on graph kernels , 2019, Applied Network Science.

[16]  Geoffrey E. Hinton,et al.  Visualizing Data using t-SNE , 2008 .

[17]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[18]  Rolf Niedermeier,et al.  Reflections on Multivariate Algorithmics and Problem Parameterization , 2010, STACS.

[19]  Mirco Musolesi,et al.  Spatio-temporal networks: reachability, centrality and robustness , 2015, Royal Society Open Science.

[20]  Vitaly Shmatikov,et al.  De-anonymizing Social Networks , 2009, 2009 30th IEEE Symposium on Security and Privacy.

[21]  Evaggelia Pitoura,et al.  Diffusion Maximization in Evolving Social Networks , 2015, COSN.

[22]  Alain Barrat,et al.  Can co-location be used as a proxy for face-to-face contacts? , 2017, EPJ Data Science.

[23]  Amir Abboud,et al.  Tight Hardness Results for LCS and Other Sequence Similarity Measures , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[24]  Ciro Cattuto,et al.  High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School , 2011, PloS one.

[25]  Aristides Gionis,et al.  Mining Temporal Networks , 2019, KDD.

[26]  Karima Benatchba,et al.  Tracking community evolution in social networks: A survey , 2019, Inf. Process. Manag..

[27]  Arnaud Casteigts,et al.  The Computational Complexity of Finding Temporal Paths under Waiting Time Constraints , 2019, ArXiv.

[28]  Richard J. Lipton,et al.  On the complexity of SAT , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[29]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[30]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[31]  William Kuszmaul,et al.  Dynamic Time Warping in Strongly Subquadratic Time: Algorithms for the Low-Distance Regime and Approximate Evaluation , 2019, ICALP.

[32]  S. Chiba,et al.  Dynamic programming algorithm optimization for spoken word recognition , 1978 .

[33]  George Karypis,et al.  Algorithms for Mining the Coevolving Relational Motifs in Dynamic Networks , 2015, ACM Trans. Knowl. Discov. Data.

[34]  Nils M. Kriege,et al.  On Valid Optimal Assignment Kernels and Applications to Graph Classification , 2016, NIPS.

[35]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[36]  Vincent Froese,et al.  Fast Exact Dynamic Time Warping on Run-Length Encoded Time Series , 2019, ArXiv.

[37]  Y.-Y. Liu,et al.  The fundamental advantages of temporal networks , 2016, Science.

[38]  Rolf Niedermeier,et al.  The Complexity of Finding Small Separators in Temporal Graphs , 2017, MFCS.

[39]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[40]  Rolf Niedermeier,et al.  Comparing temporal graphs using dynamic time warping , 2018, Social Network Analysis and Mining.

[41]  Anthony K. H. Tung,et al.  Comparing Stars: On Approximating Graph Edit Distance , 2009, Proc. VLDB Endow..

[42]  Aristides Gionis,et al.  Reconstructing an Epidemic Over Time , 2016, KDD.

[43]  Michael R. Fellows,et al.  Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity , 2013, Eur. J. Comb..

[44]  Rolf Niedermeier,et al.  Multistage s–t Path: Confronting Similarity with Dissimilarity , 2020, Algorithmica.

[45]  V. Gemmetto,et al.  Mitigation of infectious disease at school: targeted class closure vs school closure , 2014, BMC Infectious Diseases.

[46]  Stratis Ioannidis,et al.  A Family of Tractable Graph Distances , 2018, SDM.

[47]  Junjie Wu,et al.  Embedding Temporal Network via Neighborhood Formation , 2018, KDD.

[48]  Kaspar Riesen,et al.  Structural Pattern Recognition with Graph Edit Distance , 2016, Advances in Computer Vision and Pattern Recognition.

[49]  Andreas Zell,et al.  Optimal assignment kernels for attributed molecular graphs , 2005, ICML.

[50]  Thomas Gärtner,et al.  A survey of kernels for structured data , 2003, SKDD.

[51]  Tijana Milenkovic,et al.  Alignment of dynamic networks , 2017, Bioinform..

[52]  Charu C. Aggarwal,et al.  Modeling Co-Evolution Across Multiple Networks , 2018, SDM.

[53]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[54]  Rolf Niedermeier,et al.  Multistage Problems on a Global Budget , 2019, ArXiv.