Finding and tracking local communities by approximating derivatives in networks

  • M. Amin RigiEmail author
  • Irene Moser
  • M. Mehdi Farhangi
  • Chengfei Lui


Since various complex systems are represented by networks, detecting and tracking local communities has become a crucial task nowadays. Local community detection methods are getting much attention because they can address large networks. One famous class of local community detection is to find communities around a seed node. In this research, a novel local community detection method, inspired by geometric active contours, is proposed for finding a community surrounding an initial seed. While most of real world networks are dynamic and the majority of local community detection cannot tackle dynamic networks, the proposed model has the ability to track a local community in a dynamic network. The proposed model introduces and uses the derivative-based concepts curvature and gradient of the boundary of a connected sub-graph in networks. Then, a velocity function based on curvature and gradient is proposed to determine if the boundary of a community should evolve to include a neighbouring candidate. Approximating derivatives in discrete Euclidean space has a long history. However, compared to Euclidean space, graphs follow a non-uniform space in which the dimensionality, given by the the fluctuation in degrees of nodes, fluctuates from one node to another. This complexity complicates the approximation of derivatives which are needed for defining the curvature and gradient of a node in the boundary of a community. A new framework to approximate derivatives in graphs is proposed for such a purpose. For finding local communities, benchmarking our method against two recent methods indicates that it is capable of finding communities with equal or better conductance; and, for tracking dynamic local communities, benchmarking of the proposed method against ground-truth dataset shows a noticeable level of accuracy.


Derivative Graph Dynamic Differential geometry Local community detection Active contours 



  1. 1.
    Ahn, Y-Y, Bagrow, J.P., Lehmann, S.: Link communities reveal multiscale complexity in networks. Nature 466(7307), 761–764 (2010)CrossRefGoogle Scholar
  2. 2.
    Andersen, R., Lang, K.J.: Communities from seed sets. In: Proceedings of the 15th international conference on World Wide Web, pp 223–232. ACM (2006)Google Scholar
  3. 3.
    Bagrow, J.P., Bollt, E.M.: Local method for detecting communities. Phys. Rev. E 72(4), 046108 (2005)CrossRefGoogle Scholar
  4. 4.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56 (1-3), 89–113 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brandes, U., Delling, D., Gaertler, M., Görke, R, Hoefer, M., Nikoloski, Z., Wagner, D.: On modularity-np-completeness and beyond. Citeseer (2006)Google Scholar
  6. 6.
    Canu, M., Lesot, M-J, d’Allonnes, A.R.: Vertex-centred method to detect communities in evolving networks. In: International workshop on complex networks and their applications, pp 275–286. Springer (2016)Google Scholar
  7. 7.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)zbMATHCrossRefGoogle Scholar
  8. 8.
    Chan, T.F., Vese, L., et al.: Active contours without edges. IEEE Trans. Image. Process. 10(2), 266–277 (2001)zbMATHCrossRefGoogle Scholar
  9. 9.
    Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. In: 44th Annual IEEE symposium on foundations of computer science, 2003. Proceedings, pp 524–533. IEEE (2003)Google Scholar
  10. 10.
    Chen, J., Yuan, B.: Detecting functional modules in the yeast protein–protein interaction network. Bioinformatics 22(18), 2283–2290 (2006)CrossRefGoogle Scholar
  11. 11.
    Chen, J., Zaiane, O.R., Goebel, R.: Detecting communities in large networks by iterative local expansion. In: International conference on computational aspects of social networks, 2009. CASON’09, pp 105–112. IEEE (2009)Google Scholar
  12. 12.
    Clauset, A.: Finding local community structure in networks. Phys. Rev. E 72 (2), 026132 (2005)CrossRefGoogle Scholar
  13. 13.
    Clauset, A., Newman, M.E., Moore, C.: Finding community structure in very large networks. Phys. Rev. E 70(6), 066111 (2004)CrossRefGoogle Scholar
  14. 14.
    Coleman, J.S., et al.: Introduction to mathematical sociology. London Free Press, Glencoe (1964)Google Scholar
  15. 15.
    Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen differenzengleichungen der mathematischen physik. Math. Ann. 100(1), 32–74 (1928)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Danon, L., Diaz-Guilera, A., Duch, J., Arenas, A.: Comparing community structure identification. J. Stat. Mech:. Theory. Exper. 2005(09), P09008 (2005)zbMATHCrossRefGoogle Scholar
  17. 17.
    Diao, P., Guillot, D., Khare, A., Rajaratnam, B.: Differential calculus on graphon space. J. Comb. Theory. Series. A 133, 183–227 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Du, H., Feldman, M.W., Li, S., Jin, X.: An algorithm for detecting community structure of social networks based on prior knowledge and modularity. Complexity 12 (3), 53–60 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Falkowski, T.: Community analysis in dynamic social networks. Sierke (2009)Google Scholar
  20. 20.
    Farhangi, M.M., Frigui, H., Bert, R., Amini, A.A.: Incorporating shape prior into active contours with a sparse linear combination of training shapes: application to corpus callosum segmentation. In: 2016 IEEE 38th Annual international conference of the engineering in medicine and biology society (EMBC), pp 6449–6452. IEEE (2016)Google Scholar
  21. 21.
    Farhangi, M.M., Frigui, H., Seow, A., Amini, A.A.: 3-d active contour segmentation based on sparse linear combination of training shapes (scots). IEEE Trans. Med. Imaging. 36(11), 2239–2249 (2017)CrossRefGoogle Scholar
  22. 22.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3), 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Friedman, J., Tillich, J-P: Calculus on graphs. arXiv preprint cs/0408028 (2004)Google Scholar
  24. 24.
    Golub, G.H., Van Loan, C.F.: Matrix computations, vol 3. JHU Press (2012)Google Scholar
  25. 25.
    Görke, R, Kluge, R., Schumm, A., Staudt, C., Wagner, D.: An efficient generator for clustered dynamic random networks. In: Design and analysis of algorithms - first mediterranean conference on algorithms, MedAlg 2012, Kibbutz Ein Gedi, Israel, December 3-5, 2012. Proceedings, volume 7659 of lecture notes in computer science (LNCS), pp 219–233. Springer International Publishing, Switzerland (2012)Google Scholar
  26. 26.
    Hu, Y., Yang, B., Lv, C.: A local dynamic method for tracking communities and their evolution in dynamic networks. Knowl-Based. Syst. 110, 176–190 (2016)CrossRefGoogle Scholar
  27. 27.
    Jeub, L.G., Balachandran, P., Porter, M.A., Mucha, P.J., Mahoney, M.W.: Think locally, act locally: detection of small, medium-sized, and large communities in large networks. Phys. Rev. E 91(1), 012821 (2015)CrossRefGoogle Scholar
  28. 28.
    Johnson, S.C.: Hierarchical clustering schemes. Psychometrika 32(3), 241–254 (1967)zbMATHCrossRefGoogle Scholar
  29. 29.
    Khorasgani, R.R., Chen, J., Zaiane, O.R.: Top leaders community detection approach in information networks. In: 4th SNA-KDD workshop on social network mining and analysis (2010)Google Scholar
  30. 30.
    Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. J. ACM 46(5), 604–632 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Kottak CP: Cultural anthropology: appreciating cultural diversity. McGraw-Hill (2011)Google Scholar
  32. 32.
    Krishnamurthy, B., Wang, J.: On network-aware clustering of Web clients. ACM SIGCOMM. Comput. Commun. Rev. 30(4), 97–110 (2000)CrossRefGoogle Scholar
  33. 33.
    Lancichinetti, A., Fortunato, S.: Community detection algorithms: a comparative analysis. Phys. Rev. E 80(5), 056117 (2009)CrossRefGoogle Scholar
  34. 34.
    Lancichinetti, A., Fortunato, S., Kertész, J: Detecting the overlapping and hierarchical community structure in complex networks. J. Phys. 11(3), 033015 (2009)Google Scholar
  35. 35.
    Lawson, C.L., Hanson, R.J.: Solving least squares problems, vol 15. SIAM (1995)Google Scholar
  36. 36.
    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graphs over time: densification laws, shrinking diameters and possible explanations. In: Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, pp 177–187. ACM (2005)Google Scholar
  37. 37.
    Leskovec, J., Lang, K.J., Dasgupta, A., Mahoney, M.W.: Community structure in large networks: natural cluster sizes and the absence of large well-defined clusters. Internet. Math. 6(1), 29–123 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Leskovec, J., Lang, K.J., Mahoney, M.: Empirical comparison of algorithms for network community detection. In: Proceedings of the 19th international conference on World wide Web, pp 631–640. ACM (2010)Google Scholar
  39. 39.
    Mahoney, M.W., Orecchia, L., Vishnoi, N.K.: A local spectral method for graphs: with applications to improving graph partitions and exploring data graphs locally. J. Mach. Learn. Res. 13(1), 2339–2365 (2012)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Mcauley, J., Leskovec, J.: Discovering social circles in ego networks. ACM Trans. Knowl. Discov. Data. (TKDD) 8(1), 4 (2014)Google Scholar
  41. 41.
    Ng, A.Y., Jordan, M.I., Weiss, Y., et al.: On spectral clustering: analysis and an algorithm. Adv. Neural. Inf. Process. Syst. 2, 849–856 (2002)Google Scholar
  42. 42.
    Nguyen, N.P., Dinh, T.N., Tokala, S., Thai, M.T.: Overlapping communities in dynamic networks: their detection and mobile applications. In: Proceedings of the 17th annual international conference on mobile computing and networking, pp 85–96. ACM (2011)Google Scholar
  43. 43.
    Palla, G., Derényi, I, Farkas, I., Vicsek, T.: Uncovering the overlapping community structure of complex networks in nature and society. Nature 435(7043), 814–818 (2005)CrossRefGoogle Scholar
  44. 44.
    Radicchi, F., Castellano, C., Cecconi, F., Loreto, V., Parisi, D.: Defining and identifying communities in networks. Proc. Natl. Acad. Sci. USA 101(9), 2658–2663 (2004)CrossRefGoogle Scholar
  45. 45.
    Rand, W.M.: Objective criteria for the evaluation of clustering methods. J. Am. Stat. Assoc. 66(336), 846–850 (1971)CrossRefGoogle Scholar
  46. 46.
    Rives, A.W., Galitski, T.: Modular organization of cellular networks. Proc. Natl. Acad. Sci. 100(3), 1128–1133 (2003)CrossRefGoogle Scholar
  47. 47.
    Rigi, M.A., Moser, I., Rigi, S., Liu, C.: Re-imaginig the networks: detecting local communities in networks by approximating derivatives in graph space. In: Proceedings of the 2017 IEEE/ACM international conference on advances in social networks analysis and mining 2017, ASONAM ’17, pp 974–981. ACM, New York (2017)Google Scholar
  48. 48.
    Samie, M.E., Hamzeh, A.: Community detection in dynamic social networks: a local evolutionary approach. J Inf Sci, 0165551516657717 (2016)Google Scholar
  49. 49.
    Schaeffer, S.E.: Graph clustering. Comput. Sci. Rev. 1(1), 27–64 (2007)zbMATHCrossRefGoogle Scholar
  50. 50.
    Serrano, M., Boguñá, M, Pastor-Satorras, R., Vespignani, A.: Large scale structure and dynamics of complex networks: from information technology to finance and natural sciences (2007)Google Scholar
  51. 51.
    Shang, J., Liu, L., Li, X., Xie, F., Wu, C.: Targeted revision: a learning-based approach for incremental community detection in dynamic networks. Physica. A:. Stat. Mech. Appl. 443(Supplement C), 70–85 (2016)CrossRefGoogle Scholar
  52. 52.
    Solomon, J.: Pde approaches to graph analysis. arXiv:1505.00185 (2015)
  53. 53.
    Spiliopoulou, M., Ntoutsi, I., Theodoridis, Y., Schult, R.: Monic: modeling and monitoring cluster transitions. In: Proceedings of the 12th ACM SIGKDD international conference on knowledge discovery and data mining, pp 706–711. ACM (2006)Google Scholar
  54. 54.
    Sun, J., Faloutsos, C., Papadimitriou, S., Yu, P.S.: Graphscope: parameter-free mining of large time-evolving graphs. In: Proceedings of the 13th ACM SIGKDD international conference on knowledge discovery and data mining, pp 687–696. ACM (2007)Google Scholar
  55. 55.
    Takaffoli, M., Fagnan, J., Sangi, F., Zaïane, OR: Tracking changes in dynamic information networks. In: 2011 International conference on computational aspects of social networks (CASoN), pp 94–101. IEEE (2011)Google Scholar
  56. 56.
    Takaffoli, M., Rabbany, R., Zaïane, OR: Incremental local community identification in dynamic social networks. In: Proceedings of the 2013 IEEE/ACM international conference on advances in social networks analysis and mining, pp 90–94. ACM (2013)Google Scholar
  57. 57.
    Tong, H., Papadimitriou, S., Sun, J., Yu, P.S., Faloutsos, C.: Colibri: fast mining of large static and dynamic graphs. In: Proceedings of the 14th ACM SIGKDD international conference on knowledge discovery and data mining, pp 686–694. ACM (2008)Google Scholar
  58. 58.
    Toyoda, M., Kitsuregawa, M.: Creating a Web community chart for navigating related communities. In: Proceedings of the 12th ACM conference on hypertext and hypermedia, HYPERTEXT ’01, pp 103–112. ACM, New York (2001)Google Scholar
  59. 59.
    Traud, A.L., Mucha, P.J., Porter, M.A.: Social structure of facebook networks. Physica. A:. Stat. Mech. Appl. 391(16), 4165–4180 (2012)CrossRefGoogle Scholar
  60. 60.
    Xu, T., Zhang, Z., Yu, P.S., Long, B.: Generative models for evolutionary clustering. ACM Trans. Knowl. Discov. Data. (TKDD) 6(2), 7 (2012)Google Scholar
  61. 61.
    Yang, T., Chi, Y., Zhu, S., Gong, Y., Jin, R.: Detecting communities and their evolutions in dynamic social networks—a bayesian approach. Mach. Learn. 82(2), 157–189 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Yoshida, T.: Toward finding hidden communities based on user profile. J. Intell. Inf. Syst. 40(2), 189–209 (2013)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Swinburne University of TechnologyMelbourneAustralia
  2. 2.University of LouisvilleLouisvilleUSA

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