Skip to main content
SpringerLink
Log in

Transform-based graph topology similarity metrics

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

Graph signal processing has recently emerged as a field with applications across a broad spectrum of fields including brain connectivity networks, logistics and supply chains, social media, computational aesthetics, and transportation networks. In this paradigm, signal processing methodologies are applied to the adjacency matrix, seen as a two-dimensional signal. Fundamental operations of this type include graph sampling, the graph Laplace transform, and graph spectrum estimation. In this context, topology similarity metrics allow meaningful and efficient comparisons between pairs of graphs or along evolving graph sequences. In turn, such metrics can be the algorithmic cornerstone of graph clustering schemes. Major advantages of relying on existing signal processing kernels include parallelism, scalability, and numerical stability. This work presents a scheme for training a tensor stack network to estimate the topological correlation coefficient between two graph adjacency matrices compressed with the two-dimensional discrete cosine transform, augmenting thus the indirect decompression with knowledge stored in the network. The results from three benchmark graph sequences are encouraging in terms of mean square error and complexity especially for graph sequences. An additional key point is the independence of the proposed method from the underlying domain semantics. This is primarily achieved by focusing on higher-order structural graph patterns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Ahmed N, Natarajan T, Rao KR (1974) Discrete cosine transform. IEEE Trans Comput 100(1):90–93

    Article  MathSciNet  Google Scholar 

  2. Batagelj V, Mrvar A (2000) Some analyses of Erdos collaboration graph. Soc Netw 22(2):173–186

    Article  MathSciNet  Google Scholar 

  3. Besard T, Foket C, De Sutter B (2018) Effective extensible programming: unleashing Julia on GPUs. IEEE Trans Parallel Distrib Syst 30(4):827–841

    Article  Google Scholar 

  4. Bezanson J, Edelman A, Karpinski S, Shah VB (2017) Julia: a fresh approach to numerical computing. SIAM Rev 59(1):65–98

    Article  MathSciNet  Google Scholar 

  5. Bezanson J, Chen J, Chung B, Karpinski S, Shah VB, Vitek J, Zoubritzky L (2018) Julia: dynamism and performance reconciled by design. ACM Conf Program Lang 2(OOPSLA):1–23

  6. Buccini A, Pasha M, Reichel L (2020) Generalized singular value decomposition with iterated Tikhonov regularization. J Comput Appl Math. https://doi.org/10.1016/j.cam.2019.05.024

    Article  MathSciNet  MATH  Google Scholar 

  7. Byun SW, Son HS, Lee SP (2019) Fast and robust watermarking method based on DCT specific location. IEEE Access 7:100706–100718

    Article  Google Scholar 

  8. Chen S, Varma R, Sandryhaila A, Kovačević J (2015) Discrete signal processing on graphs: sampling theory? IEEE Trans Signal Process 63(24):6510–6523

    Article  MathSciNet  Google Scholar 

  9. Drakopoulos G, Kafeza E (2020) One dimensional cross-correlation methods for deterministic and stochastic graph signals with a Twitter application in Julia. SEEDA-CECNSM. https://doi.org/10.1109/SEEDA-CECNSM49515.2020.9221815

    Article  Google Scholar 

  10. Drakopoulos G, Mylonas P (2020) Evaluating graph resilience with tensor stack networks: a keras implementation. NCAA 32(9):4161–4176. https://doi.org/10.1007/s00521-020-04790-1

    Article  Google Scholar 

  11. Drakopoulos G, Mylonas P, Sioutas S (2019) A case of adaptive nonlinear system identification with third order tensors in TensorFlow. INISTA. https://doi.org/10.1109/INISTA.2019.8778406

    Article  Google Scholar 

  12. Drakopoulos G, Giannoukou I, Mylonas P, Sioutas S (2020) On tensor distances for self organizing maps: clustering cognitive tasks. DEXA. Springer, Cham, pp 195–210

    Google Scholar 

  13. Drakopoulos G, Stathopoulou F, Kanavos A, Paraskevas M, Tzimas G, Mylonas P, Iliadis L (2020) A genetic algorithm for spatiosocial tensor clustering. EVOS 11(3):491–501. https://doi.org/10.1007/s12530-019-09274-9

    Article  Google Scholar 

  14. Ekambaram VN, Fanti GC, Ayazifar B, Ramchandran K (2013) Circulant structures and graph signal processing. In: ICIP, pp 834–838. IEEE

  15. Feig E, Winograd S (1992) Fast algorithms for the discrete cosine transform. IEEE Trans Signal Process 40(9):2174–2193

    Article  Google Scholar 

  16. Gama F, Marques AG, Leus G, Ribeiro A (2018) Convolutional neural network architectures for signals supported on graphs. IEEE Trans Signal Process 67(4):1034–1049

    Article  MathSciNet  Google Scholar 

  17. Gori M, Monfardini G, Scarselli F (2005) A new model for learning in graph domains. IJCNN 2:729–734

    Google Scholar 

  18. Huang W, Bolton TA, Medaglia JD, Bassett DS, Ribeiro A, Van De Ville D (2018) A graph signal processing perspective on functional brain imaging. Proceedings of the IEEE 106(5):868–885

    Article  Google Scholar 

  19. Jeong H, Mason S, Barabasi A, Oltvai Z (2001) Lethality and centrality in protein networks. arXiv preprint cond-mat/0105306

  20. Jia Z, Yang Y (2020) A joint bidiagonalization based iterative algorithm for large scale general-form Tikhonov regularization. Appl Numer Math 157:159–177

    Article  MathSciNet  Google Scholar 

  21. Khamparia A, Gupta D, Nguyen NG, Khanna A, Pandey B, Tiwari P (2019) Sound classification using convolutional neural network and tensor deep stacking network. IEEE Access 7:7717–7727

    Article  Google Scholar 

  22. Kisel’ák J, Lu Y, Švihra J, Szépe P, Stehlik M (2021) SPOCU: scaled polynomial constant unit activation function. NCAA 33(8):3385–3401

    Article  Google Scholar 

  23. Knopp T, Szwargulski P, Griese F, Grosser M, Boberg M, Möddel M (2019) MPIReco. jl: Julia package for image reconstruction in MPI. Int J Magn Part Imag. https://doi.org/10.18416/IJMPI.2019.1907001

  24. Lattner C (2008) LLVM and Clang: next generation compiler technology. The BSD conference, BSD Foundation 5:1–33

    Google Scholar 

  25. Lattner C, Adve V (2004) LLVM: a compilation framework for lifelong program analysis & transformation. In: CGO, pp 75–86. IEEE

  26. Li Z, Chen J (2017) Robust consensus of linear feedback protocols over uncertain network graphs. IEEE Trans Autom Control 62(8):4251–4258

    Article  MathSciNet  Google Scholar 

  27. Lian D, Xie X, Chen E (2019) Discrete matrix factorization and extension for fast item recommendation. IEEE Trans Knowl Data Eng 33:1919–1933

    Google Scholar 

  28. Liang L, Xu J, Deng L, Yan M, Hu X, Zhang Z, Li G, Xie Y (2021) Fast search of the optimal contraction sequence in tensor networks. IEEE J Select Topics Signal Process 15:574–586

    Article  Google Scholar 

  29. Lu C, Feng J, Chen Y, Liu W, Lin Z, Yan S (2019) Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Trans Pattern Anal Mach Intell 42(4):925–938

    Article  Google Scholar 

  30. Lubin M, Dunning I (2015) Computing in operations research using Julia. INFORMS Journal on Computing 27(2):238–248

    Article  MathSciNet  Google Scholar 

  31. Micheli A (2009) Neural network for graphs: a contextual constructive approach. IEEE Trans Neural Netw 20(3):498–511

    Article  MathSciNet  Google Scholar 

  32. Mogensen PK, Riseth AN (2018) Optim: a mathematical optimization package for Julia. J Open Source Softw. https://doi.org/10.21105/joss.00615

    Article  Google Scholar 

  33. Ortega A, Frossard P, Kovačević J, Moura JM, Vandergheynst P (2018) Graph signal processing: overview, challenges, and applications. Proc IEEE 106(5):808–828

    Article  Google Scholar 

  34. Pang CY, Zhou RG, Hu BQ, Hu W, El-Rafei A (2019) Signal and image compression using quantum discrete cosine transform. Inf Sci 473:121–141

    Article  MathSciNet  Google Scholar 

  35. Paramanandham N, Rajendiran K (2018) Infrared and visible image fusion using discrete cosine transform and swarm intelligence for surveillance applications. Infrared Phys Technol 88:13–22

    Article  Google Scholar 

  36. Rossi RA, Ahmed NK (2015) The network data repository with interactive graph analytics and visualization. In: AAAI, URL http://networkrepository.com

  37. Sandryhaila A, Moura JM (2014) Discrete signal processing on graphs: frequency analysis. IEEE Trans Signal Process 62(12):3042–3054

    Article  MathSciNet  Google Scholar 

  38. Shi J, Wen F, Liu T (2020) Nested MIMO radar: coarrays, tensor modeling and angle estimation. IEEE Trans Aerosp Electron Syst 57:573–585

    Google Scholar 

  39. Strang G (1999) The discrete cosine transform. SIAM Rev 41(1):135–147

    Article  MathSciNet  Google Scholar 

  40. Tremblay N, Amblard PO, Barthelmé S (2017) Graph sampling with determinantal processes. In: EUSIPCO, pp 1674–1678. IEEE

  41. Verma J, Gupta S, Mukherjee D, Chakraborty T (2019) Heterogeneous edge embedding for friend recommendation. ECIR. Springer, Cham, pp 172–179

    Google Scholar 

  42. Wang M, Shang X (2020) A fast image fusion with discrete cosine transform. IEEE Signal Process Lett 27:990–994

    Article  Google Scholar 

  43. Wang SJ, Yang J, Zhang N, Zhou CG (2011) Tensor discriminant color space for face recognition. IEEE Trans Image Process 20(9):2490–2501

    Article  MathSciNet  Google Scholar 

  44. Wang X, Che M, Wei Y (2019) Neural networks based approach solving multi-linear systems with M-tensors. Neurocomputing 351:33–42

    Article  Google Scholar 

  45. Watts DJ, Strogatz SH (1998) Collective dynamics of small-world networks. Nature 393(6684):440–442

    Article  Google Scholar 

  46. Wu Y, Cao N, Archambault D, Shen Q, Qu H, Cui W (2016) Evaluation of graph sampling: a visualization perspective. IEEE Trans Vis Comput Graphics 23(1):401–410

    Article  Google Scholar 

  47. Wu Z, Pan S, Chen F, Long G, Zhang C, Philip SY (2020) A comprehensive survey on graph neural networks. IEEE Trans Neural Netw Learn Syst 32:4–24

    Article  MathSciNet  Google Scholar 

  48. Yu D, Deng L, Seide F (2012) The deep tensor neural network with applications to large vocabulary speech recognition. IEEE Trans Audio, Speech, Lang Process 21(2):388–396

    Article  Google Scholar 

  49. Zhang C, Cai M, Zhao X, Wang D (2021) Research on case preprocessing based on deep learning. Concurr Comput Pract Exper. https://doi.org/10.1002/cpe.6214

    Article  Google Scholar 

  50. Zhou J, Cui G, Zhang Z, Yang C, Liu Z, Wang L, Li C, Sun M (2018) Graph neural networks: a review of methods and applications. arXiv preprint 81208434

Download references

Acknowledgements

This article is part of Project 451, a long-term research initiative whose primary objective is the development of novel, scalable, numerically stable, and interpretable tensor analytics.

Author information

Authors and Affiliations

  1. Department of Informatics, Ionian University, Corfu, Greece

    Georgios Drakopoulos & Phivos Mylonas

  2. College of Technology and Innovation, Zayed University, Dubai, UAE

    Eleanna Kafeza

  3. Laboratory of Mathematics and Informatics, Democritus University of Thrace, Xanthi, Greece

    Lazaros Iliadis

Authors
  1. Georgios Drakopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eleanna Kafeza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Phivos Mylonas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lazaros Iliadis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgios Drakopoulos.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drakopoulos, G., Kafeza, E., Mylonas, P. et al. Transform-based graph topology similarity metrics. Neural Comput & Applic 33, 16363–16375 (2021). https://doi.org/10.1007/s00521-021-06235-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-021-06235-9

Keywords

Mathematics subject classification

Search

Navigation