QSIM: A novel approach to node proximity estimation based on Discrete-time quantum walk

Abstract

Node proximity estimation studies structural similarity between nodes and is the key issue of network analysis. It can exist as the node recommendation task and is a fundamental basis of other graph mining techniques. Although Discrete-time quantum walk (DTQW), a promising new technique with distinctive characters, is widely used in many graph mining problems such as graph isomorphism and graph kernel, there are only a few works estimating proximity via DTQW, limiting the further application of DTQW in graph mining. In this paper, we study the capability of DTQW for proximity estimation and propose QSIM to estimate node proximity by DTQW. By analyzing the diffusion process of biased walks, we discover two influential effects that are beneficial to proximity estimation. The Diminishing Effect shows that a node close to the starting node can generally have a high average probability during the diffusion process, which serves as the basis of QSIM. The Returning Effect shows the probability has a tendency to stay around the starting node during the diffusion, which enhances the capability for mining local information especially in densely-connected structures. Benefited from the two effects, QSIM faithfully reveals node proximity and comprehensively unifies different kinds of node proximity. QSIM is the first mature quantum-walk-based method for proximity estimation. Extensive experiments validate the effectiveness of QSIM and show that QSIM outperforms state-of-the-art methods in the node recommendation task, significantly surpassing Refex, Node2vec, and Role2vec, by up to 1094.2% in the first-order node proximity and 18.8% in the second-order node proximity.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Refex is available on https://github.com/randomsurfer/refex, Node2vec is available on https://github.com/aditya-grover/node2vec, Role2vec is available on https://github.com/benedekrozemberczki/role2vec.

References

  1. 1.

    Ahmed NK, Rossi RA, Lee JB, Kong X, Willke TL, Zhou R, Eldardiry H (2018) Learning role-based graph embeddings. Stat 1050:7

    Google Scholar 

  2. 2.

    Bai L, Rossi L, Cui L, Zhang Z, Ren P, Bai X, Hancock E (2017) Quantum kernels for unattributed graphs using discrete-time quantum walks. Pattern Recogn Lett 87:96–103

    Article  Google Scholar 

  3. 3.

    Cai H, Zheng VW, Chang KC (2018) A comprehensive survey of graph embedding: problems, techniques, and applications. IEEE Trans Knowl Data Eng 30(9):1616–1637

    Article  Google Scholar 

  4. 4.

    Childs AM (2010) On the relationship between continuous-and discrete-time quantum walk. Commun Math Phys 294(2):581–603

    MathSciNet  Article  Google Scholar 

  5. 5.

    Childs AM, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman DA (2003) Exponential algorithmic speedup by a quantum walk. In: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. ACM, pp 59–68

  6. 6.

    Cross R, Parker A, Christensen CM, Anthony SD, Roth EA (2004) The hidden power of social networks. Journal of Applied Management & Entrepreneurship 9(Oct)

  7. 7.

    Douglas B, Wang J (2008) A classical approach to the graph isomorphism problem using quantum walks, vol 41

  8. 8.

    Emms D, Wilson RC, Hancock ER (2009) Graph matching using the interference of discrete-time quantum walks. Image Vis Comput 27(7):934–949

    Article  Google Scholar 

  9. 9.

    Ribeiro LFR, Savarese PHP, Figueiredo DR (2017) Struc2vec: Learning node representations from structural identity. In: International ACM conference on knowledge discovery and data mining (KDD). ACM, pp 385–394

  10. 10.

    Freeman LC (1978) Centrality in social networks conceptual clarification. Soc Netw 1(3):215–239

    Article  Google Scholar 

  11. 11.

    Fujiwara Y, Nakatsuji M, Shiokawa H, Mishima T, Onizuka M (2013) Efficient ad-hoc search for personalized pagerank. In: ACM SIGMOD International conference on management of data. ACM, pp 445–456

  12. 12.

    Girvan M, Newman ME (2002) Community structure in social and biological networks. Proc Natl Acad Sci USA 99(12):7821–7826

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gleiser PM, Danon L (2003) Community structure in jazz. Adv Compl Syst 6(04):565–573

    Article  Google Scholar 

  14. 14.

    Grover A (2016) Leskovec, J.: node2vec: Scalable feature learning for networks. In: International ACM conference on knowledge discovery and data mining (KDD). ACM, pp 855–864

  15. 15.

    Grover LK (1996) A fast quantum mechanical algorithm for database search. In: Acm symposium on theory of computing, pp 212–219

  16. 16.

    Guimera R, Danon L, Diaz-Guilera A, Giralt F, Arenas A (2003) Self-similar community structure in a network of human interactions. Phys Rev E 68(6):065103

  17. 17.

    Henderson K, Gallagher B, Li L, Akoglu L, Eliassi-Rad T, Tong H, Faloutsos C (2011) It’s who you know: Graph mining using recursive structural features. In: International ACM conference on knowledge discovery and data mining (KDD). ACM, pp 663–671

  18. 18.

    Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: Densification and shrinking diameters. ACM Trans Knowl Discov Data (TKDD) 1(1):2

    Article  Google Scholar 

  19. 19.

    Lusseau D, Schneider K, Boisseau OJ, Haase P, Slooten E, Dawson SM (2003) The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations. Behav Ecol Sociobiol 54(4):396–405

    Article  Google Scholar 

  20. 20.

    Mahasinghe A, Izaac JA, Wang JB, Wijerathna JK (2015) Phase-modified ctqw unable to distinguish strongly regular graphs efficiently. J Phys A: Math Theor 48(26):265301

  21. 21.

    Moody J (2001) Peer influence groups: identifying dense clusters in large networks. Soc Netw 23 (4):261–283

    Article  Google Scholar 

  22. 22.

    Newman ME (2006) Modularity and community structure in networks. Proc Natl Acad Sci U S A 103(23):8577–8582

    Article  Google Scholar 

  23. 23.

    Perozzi B, Alrfou R, Skiena S (2014) Deepwalk: online learning of social representations. In: International ACM conference on knowledge discovery and data mining (KDD). ACM, pp 701–710

  24. 24.

    Porter MA, Onnela JP, Mucha PJ (2009) Communities in networks. Not Am Math Soc 56(9):4294–4303

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Rohde PP, Fedrizzi A, Ralph TC (2012) Entanglement dynamics and quasi-periodicity in discrete quantum walks. J Mod Opt 59(8):710–720

    Article  Google Scholar 

  26. 26.

    Sailer LD (1984) Proximity, sociality, and observation: The definition of social groups. Am Anthropol 86(1):91–98

    Article  Google Scholar 

  27. 27.

    Santha M (2008) Quantum walk based search algorithms. In: International conference on theory and applications of models of computation. Springer, pp 31–46

  28. 28.

    Tang L, Liu H (2009) Relational learning via latent social dimensions. In: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp 817–826

  29. 29.

    Tsomokos DI (2010) Community detection in complex networks with quantum random walks. arXiv:1012.2405

  30. 30.

    Tsomokos DI (2011) Quantum walks on complex networks with connection instabilities and community structure. Phys Rev A 83(5):052315

Download references

Acknowledgements

This work is supported by National High-level Personnel for Defense Technology Program (2017-JCJQ-ZQ-013), NSF 61902405, and the China Scholarship Council (CSC Student ID 201903170136). This work is partially done during my research visit to School of Computing, National University of Singapore.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xin Wang.

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

Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Lu, K., Zhang, Y. et al. QSIM: A novel approach to node proximity estimation based on Discrete-time quantum walk. Appl Intell 51, 2574–2588 (2021). https://doi.org/10.1007/s10489-020-01970-3

Download citation

Keywords

  • Quantum walk
  • Discrete-time quantum walk
  • Node proximity estimation
  • Network analysis