4×4 Knight’s Graph Analysis by Modularity A Knight Graph Application

Abstract

Modularity is a well-known technic to analyze graphs of datasets. The modularity divides the network into meaningful clusters. The clusters present the existence of shared properties between the nodes in the same cluster. In the present study, we analyze 4×4 knight graph by modularity to investigate 4-KCP solutions. Our investigation is completed for resolutions from 0.2 to 3.2. The maximum modularity score is 0.417 found (for resolution = 0.6, 0.8, 1, and 1.4). Moreover, resolution 1.4 is the optimum resolution to find some solutions of 4-KCP. Also, the analyses show resolution 0.2 is the best resolution to find all solutions of 4-KCP.

Keywords

• Knight graph
• Modularity
• 4-KCP

References

1. S. Bai, G. Zhu, and J. Huang, "An Intelligent Algorithm for the (1,2,2)-Generalized Knight's Tour Problem," in 2013 Ninth International Conference on Computational Intelligence and Security, 2013, pp. 583-588.
2. H. Jian and B. Sen, "An Efficient Algorithm for the Generalized (1,k)-Knight's Tours Problem," in 2009 First International Workshop on Education Technology and Computer Science, 2009, vol. 1, pp. 697-701.
3. S. Bai, X. Liao, X. Qu, and Y. Liu, "Generalized Knight's Tour Problem and Its Solutions Algorithm," in 2006 International Conference on Computational Intelligence and Security, 2006, vol. 1, pp. 570-573.
4. A. Philip, "A Generalized Pseudo-Knight?s Tour Algorithm for Encryption of an Image," IEEE Potentials, vol. 32, no. 6, pp. 10-16, 2013.
5. J. Kumar and S. Nirmala, "Securing the contents of document images using knight moves and genetic approach," in 2015 International Conference on Advances in Computing, Communications and Informatics (ICACCI), 2015, pp. 1091-1095.
6. J. Delei, B. Sen, and D. Wenming, "An Image Encryption Algorithm Based on Knight's Tour and Slip Encryption-Filter," in 2008 International Conference on Computer Science and Software Engineering, 2008, vol. 1, pp. 251-255.
7. D. C. Fisher, "On the n x n Knight Cover Problem," Ars Comb., vol. 69, / 2003.
8. F. Rubin, "Improved Knight Coverings," Ars Comb., vol. 69, / 2003.

9. F. Rubin, "Knight Covers for the 50x50 Chessboard," presented at the Mathfest 2004, Providence RI, 2004.
10. F. Rubin, "A Family of Efficient Knight Covering Patterns," Journal of Recreational Mathematics, Article vol. 33, no. 3, pp. 165-175, 2005.
11. F. Rubin, "An Improved Method for Finding Knight Covers," Ars Comb., vol. 82, / 2007.
12. B. Lemaire, "Knights Covers on NxN Chessboards," J. Recr. Math., vol. 31, pp. 87-99, 2003.
13. A. H. Jackson and R. P. Pargas, "Solutions to the NxN Knights Covering Problem," J. Recr. Math. , vol. 23, pp. 255-267, 1991.
14. F. Wei, "Research on Knight Covering Based on Breadth First Search Algorithm," (in English), APPLIED MECHANICS AND MATERIALS, vol. 686, pp. 377-380, 2014.
15. S. Güldal, M. Lipscomb, and M. M. Tanik, "Solving Knights Covering Problem: Backtracking, Permutation, Bipartite Graph, and Independent Set," presented at the Nineteenth Annual Early Career Technical Conference, Birmingham, Alabama USA, 2019
16. S. Güldal, M. M. Tanik, and M. M. Lipscomb, "Solving Knights Covering Problem by a Hybrid Algorithm," presented at the IEEE SouthEastConn, Huntsville, Alabama, April 11 - 14 2019,
17. S. Güldal, "Connectives of Knights Covering Problem By Girvan-Newman Clustering," presented at the SDPS 2019 Workshop, Madrid, Spain, 25-26, November 2019,
18. M. Gençer, "Sosyal Ağ Analizi Yöntemlerine Bir Bakış," Yildiz Social Science Review, vol. 3, pp. 19-34, 12/15 2017.
19. P. Nerurkar, M. Chandane, and S. Bhirud, "Understanding attribute and social circle correlation in social networks," Turkish Journal of Electrical Engineering and Computer Sciences, vol. 27, pp. 1228-1242, 03/22 2019.
20. N. Almolhem, Y. Rahal, and M. Dakkak, "Social network analysis in Telecom data," Journal of Big Data, vol. 6, 12/01 2019.
21. "Analysis of the Application of Artificial Intelligence in Computer Networks Technology," IOP Conference Series: Materials Science and Engineering, vol. 750, p. 012097, 03/24 2020.
22. C. Engström and S. Silvestrov, "PageRank for networks, graphs, and Markov chains," Theory of Probability and Mathematical Statistics, vol. 96, p. 1, 10/05 2018.
23. Y. L. Karpov, I. Volkova, A. A. Vylitok, L. Karpov, and Y. G. Smetanin, "Designing classes’ interfaces for neural network graph model," Proceedings of the Institute for System Programming of RAS, vol. 31, pp. 97-112, 10/01 2019.
24. L. Johnsen, "Graph Analysis of Word Networks," http://ceur-ws.org/Vol-2021/, vol. 2021, 01/01 2017.
25. E. Hasanah and D. Agustiningsih, Analysis of “Halal” Word in Social Media Using Text Mining and Word Networking. 2020.
26. R. Ozcelik, G. Uludoğan, S. Parlar, Ö. Bakay, O. Ergelen, and O. Yildiz, User Interface for Turkish Word Network KeNet. 2019, pp. 1-4.
27. K. Kugler, L. Mueller, A. Graber, and M. Dehmer, "Integrative Network Biology: Graph Prototyping for Co-Expression Cancer Networks," PloS one, vol. 6, p. e22843, 07/29 2011.
28. B. Eckman and P. Brown, "Graph data management for molecular and cell biology," IBM Journal of Research and Development, vol. 50, pp. 545-560, 12/01 2006.
29. V. Gadiyaram, S. Vishveshwara, and S. Vishveshwara, From Quantum Chemistry to Networks in Biology: A Graph Spectral Approach to Protein Structure Analyses. 2019.
30. S. Valverde, B. Vidiella Rocamora, R. Montañez Martínez, A. Fraile, S. Sacristán, and F. García-Arenal, "Coexistence of nestedness and modularity in host–pathogen infection networks," Nature Ecology & Evolution, pp. 1-10, 03/09 2020.
31. T. Shiino, "Phylodynamic analysis of a viral infection network," Frontiers in microbiology, vol. 3, p. 278, 07/31 2012.
32. M. E. J. Newman, "Analysis of weighted networks," Physical Review E, vol. 70, no. 5, p. 056131, 11/24/ 2004.
33. M. Bastian, S. Heymann, and M. Jacomy, "Gephi: an open source software for exploring and manipulating networks," in International AAAI Conference on Weblogs and Social Media, 2009.
34. V. D. Blondel, J.-L. Guillaume, R. Lambiotte, E. J. J. o. s. m. t. Lefebvre, and experiment, "Fast unfolding of communities in large networks," vol. 2008, no. 10, p. P10008, 2008.
35. R. Lambiotte, J.-C. Delvenne, and M. J. a. p. a. Barahona, "Laplacian dynamics and multiscale modular structure in networks," 2008.