Graph Embedding Based on Euclidean Distance Matrix and its Applications

Graph embedding converts a graph into a multi-dimensional space in which the graph structural information or graph properties are maximumly preserved. It is an effective and efficient way to provide users a deeper understanding of what is behind the data and thus can benefit a lot of useful applications. However, most graph embedding methods suffer from high computation and space costs. In this paper, we present a simple graph embedding method that directly embeds the graph into its Euclidean distance space. This method does not require the learned representations to be low dimensional, but it has several good characteristics. We find that the centrality of nodes/edges can be represented by the position of nodes or the length of edges when a graph is embedded. Besides, the edge length is closely related to the density of regions in a graph. We then apply this graph embedding method into graph analytics, such as community detection, graph compression, and wormhole detection, etc. Our evaluation shows the effectiveness and efficiency of this embedding method and contends that it yields a promising approach to graph analytics.

[1]  Sarit Agami,et al.  Modelling persistence diagrams with planar point processes, and revealing topology with bagplots , 2018, Journal of Applied and Computational Topology.

[2]  Sarit Agami.  Comparison of persistence diagrams , 2020, Commun. Stat. Simul. Comput..

[3]  Alexander J. Smola,et al.  Distributed large-scale natural graph factorization , 2013, WWW.

[4]  M. Soriano-Trigueros,et al.  Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue , 2021, Mathematics.

[5]  Daniel J. Abadi,et al.  Scalable Pattern Matching over Compressed Graphs via Dedensification , 2016, KDD.

[6]  Christos Faloutsos,et al.  Graph evolution: Densification and shrinking diameters , 2006, TKDD.

[7]  Steven Skiena,et al.  DeepWalk: online learning of social representations , 2014, KDD.

[8]  Richard M. Karp,et al.  Algorithms for graph partitioning on the planted partition model , 2001, Random Struct. Algorithms.

[9]  Kevin Chen-Chuan Chang,et al.  A Comprehensive Survey of Graph Embedding: Problems, Techniques, and Applications , 2017, IEEE Transactions on Knowledge and Data Engineering.

[10]  Mohammad Al Hasan,et al.  Methods and Applications of Network Sampling , 2016 .

[11]  Jie Tang,et al.  Inferring social ties across heterogenous networks , 2012, WSDM '12.

[12]  Fang Zhou,et al.  Compression of weighted graphs , 2011, KDD.

[13]  Jean-Loup Guillaume,et al.  Fast unfolding of communities in large networks , 2008, 0803.0476.

[14]  Mikhail Tikhomirov,et al.  On computational complexity of length embeddability of graphs , 2014, Discret. Math..

[15]  Mathias Niepert,et al.  Learning Convolutional Neural Networks for Graphs , 2016, ICML.

[16]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[17]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Adilson E. Motter,et al.  Discovering Network Structure Beyond Communities , 2011, Scientific reports.

[19]  Danai Koutra,et al.  Graph Summarization Methods and Applications: A Survey , 2016 .

[20]  J. Gower.  Properties of Euclidean and non-Euclidean distance matrices , 1985 .

[21]  Jian Pei,et al.  A Survey on Network Embedding , 2017, IEEE Transactions on Knowledge and Data Engineering.

[22]  Henry Adams,et al.  Persistence Images: A Stable Vector Representation of Persistent Homology , 2015, J. Mach. Learn. Res..

[23]  Pablo M. Gleiser,et al.  Community Structure in Jazz , 2003, Adv. Complex Syst..

[24]  Rocío González-Díaz,et al.  On the stability of persistent entropy and new summary functions for TDA , 2018, Pattern Recognit..

[25]  Mária Ercsey-Ravasz,et al.  Community detection by graph Voronoi diagrams , 2014 .

[26]  Wenwu Zhu,et al.  Structural Deep Network Embedding , 2016, KDD.

[27]  C. Lavor,et al.  On the estimation of unknown distances for a class of Euclidean distance matrix completion problems with interval data , 2020 .

[28]  Bharat K. Bhargava,et al.  Visualization of wormholes in sensor networks , 2004, WiSe '04.

[29]  Arif Mahmood,et al.  Using Geodesic Space Density Gradients for Network Community Detection , 2017, IEEE Transactions on Knowledge and Data Engineering.

[30]  Cheng Li,et al.  DeepCas: An End-to-end Predictor of Information Cascades , 2016, WWW.

[31]  Audun D. Myers,et al.  Persistent Homology of Complex Networks for Dynamic State Detection , 2019, Physical review. E.

[32]  Xin Wang,et al.  Query preserving graph compression , 2012, SIGMOD Conference.

[33]  C. Sabín.  Quantum detection of wormholes , 2017, Scientific Reports.

[34]  Abdo Y. Alfakih,et al.  On the uniqueness of Euclidean distance matrix completions , 2003 .

[35]  Jie Gao,et al.  Local connectivity tests to identify wormholes in wireless networks , 2011, MobiHoc '11.

[36]  Pierre Vandergheynst,et al.  Geometric Deep Learning: Going beyond Euclidean data , 2016, IEEE Signal Process. Mag..

[37]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Jian Pei,et al.  Asymmetric Transitivity Preserving Graph Embedding , 2016, KDD.

[39]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[40]  Jian Pei,et al.  Community Preserving Network Embedding , 2017, AAAI.

[41]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[42]  Jian Pei,et al.  Community Preserving Lossy Compression of Social Networks , 2012, 2012 IEEE 12th International Conference on Data Mining.

[43]  R. Ghrist.  Barcodes: The persistent topology of data , 2007 .

[44]  Emilie Purvine,et al.  Anomaly Detection Using Persistent Homology , 2016, 2016 Cybersecurity Symposium (CYBERSEC).

[45]  Konrad J. Swanepoel,et al.  Embedding graphs in Euclidean space , 2017, Electron. Notes Discret. Math..

[46]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[47]  Jure Leskovec,et al.  node2vec: Scalable Feature Learning for Networks , 2016, KDD.

[48]  Lijun Qian,et al.  Detection of wormhole attacks in multi-path routed wireless ad hoc networks: A statistical analysis approach , 2007, J. Netw. Comput. Appl..

[49]  Wei Lu,et al.  Deep Neural Networks for Learning Graph Representations , 2016, AAAI.