Abstract
The problem of computing distances and shortest paths between vertices in graphs is one of the fundamental issues in graph theory. It is of great importance in many different applications, for example, transportation, and social network analysis. However, efficient shortest distance algorithms are still desired in many disciplines. Basically, the majority of dense graphs have ties between the shortest distances. Therefore, we consider a different approach and introduce a new measure to solve all-pairs shortest paths for undirected and unweighted graphs. This measures the shortest distance between any two vertices by considering the length and the number of all possible paths between them. The main aim of this new approach is to break the ties between equal shortest paths SP, which can be obtained by the Breadth-first search algorithm (BFS), and distinguish meaningfully between these equal distances. Moreover, using the new measure in clustering produces higher quality results compared with SP. In our study, we apply two different clustering techniques: hierarchical clustering and K-means clustering, with four different graph models, and for a various number of clusters. We compare the results using a modularity function to check the quality of our clustering results.
Supported by Saudi Arabian Cultural Bureau in London.
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References
Aggarwal, C.C., Reddy, C.K.: Data Clustering: Algorithm and Applications. Taylor and Francis Group, London (2014)
Bonner, R.: On some clustering techniques. IBM J. Res. Dev. 8, 22–32 (1964)
Clauset, A., Newman, M.E.J., Moore, C.: Finding community structure in very large networks. Phys. Rev. E 70, 6 (2004)
Csardi, G., Nepusz, T.: The igraph software package for complex network research. InterJournal Complex Syst. (2006). http://igraph.org
Everitt, S., Landau, B.S., Leese, M., Stahl, D.: Cluster Analysis, 5th edn. Wiley, London (2011)
Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3), 75–174 (2010)
Jain, A.K.: Data clustering: 50 years beyond K-means. Pattern Recognit. 31(8), 651–666 (2010)
Leskovec, J., Krevl, A.: SNAP datasets: Stanford large network dataset collection. (2014). http://snap.stanford.edu/data/index.html
Madkour, A., Aref, G., Rehman, F., Abdur Rahman, M., Basalamah, S.: A survey of shortest-path algorithm. CoRR abs/1705.02044 (2017). http://arxiv.org/abs/1705.02044
Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 103(23), 8577–8582 (2006)
Song, S., Zhao, J.: Survey of graph clustering algorithms using amazon reviews (2014). http://snap.stanford.edu/class/cs224w-2014/projects2014
Reddy, K.R.U.K.: A survey of the all-pairs shortest paths problem and its variants in graphs. Acta Universitatis Sapientiae, Informatica 8(1), 16–40 (2016). https://doi.org/10.1515/ausi-2016-0002
Zwick, U.: Exact and approximate distances in graphs — A survey. In: auf der Heide, F.M. (ed.) ESA 2001. LNCS, vol. 2161, pp. 33–48. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44676-1_3
Blondel, V.D., Guillaume, J., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech.: Theory Exp. 2008(10), P10008 (2008)
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The first author of this manuscript is grateful to the Saudi Arabian Cultural Bureau in London for financial support.
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Almulhim, F.A., Thwaites, P.A., Taylor, C.C. (2019). A New Approach to Measuring Distances in Dense Graphs. In: Nicosia, G., Pardalos, P., Giuffrida, G., Umeton, R., Sciacca, V. (eds) Machine Learning, Optimization, and Data Science. LOD 2018. Lecture Notes in Computer Science(), vol 11331. Springer, Cham. https://doi.org/10.1007/978-3-030-13709-0_17
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