Advertisement

The core decomposition of networks: theory, algorithms and applications

  • Fragkiskos D. Malliaros
  • Christos Giatsidis
  • Apostolos N. PapadopoulosEmail author
  • Michalis Vazirgiannis
Special Issue Paper

Abstract

The core decomposition of networks has attracted significant attention due to its numerous applications in real-life problems. Simply stated, the core decomposition of a network (graph) assigns to each graph node v, an integer number c(v) (the core number), capturing how well v is connected with respect to its neighbors. This concept is strongly related to the concept of graph degeneracy, which has a long history in graph theory. Although the core decomposition concept is extremely simple, there is an enormous interest in the topic from diverse application domains, mainly because it can be used to analyze a network in a simple and concise manner by quantifying the significance of graph nodes. Therefore, there exists a respectable number of research works that either propose efficient algorithmic techniques under different settings and graph types or apply the concept to another problem or scientific area. Based on this large interest in the topic, in this survey, we perform an in-depth discussion of core decomposition, focusing mainly on: (i) the basic theory and fundamental concepts, (ii) the algorithmic techniques proposed for computing it efficiently under different settings, and (iii) the applications that can benefit significantly from it.

Keywords

Core decomposition Graph mining Graph degeneracy Graph theory Algorithms 

Notes

References

  1. 1.
    Adiga, A., Vullikanti, A.K.S.: How robust is the core of a network? In: Blockeel, H., Kersting, K., Nijssen, S., Železný, F. (eds.) Machine Learning and Knowledge Discovery in Databases, pp. 541–556. Springer, Berlin (2013)Google Scholar
  2. 2.
    Aggarwal, C.C. (ed.): Social Network Data Analytics. Springer, Berlin (2011)zbMATHGoogle Scholar
  3. 3.
    Aggarwal, C.C., Wang, H.: Managing and Mining Graph Data. Springer, Berlin (2010)zbMATHCrossRefGoogle Scholar
  4. 4.
    Akbas, E., Zhao, P.: Truss-based community search: a truss-equivalence based indexing approach. Proc. VLDB Endow. 10(11), 1298–1309 (2017)CrossRefGoogle Scholar
  5. 5.
    Al-garadi, M.A., Varathan, K.D., Ravana, S.D.: Identification of influential spreaders in online social networks using interaction weighted k-core decomposition method. Phys. A 468, 278–288 (2017)CrossRefGoogle Scholar
  6. 6.
    Alvarez-Hamelin, J., Dall’Asta, L., Barrat, A., Vespignani, A.: K-core decomposition: a tool for the visualization of large scale networks. Adv. Neural Inf. Process. Syst. 18, 04 (2005)Google Scholar
  7. 7.
    Alvarez-hamelin, J.I., Barrat, A., Vespignani, A.: Large scale networks fingerprinting and visualization using the k-core decomposition. In: NIPS’06: Advances in Neural Information Processing Systems, pp. 41–50 (2006)Google Scholar
  8. 8.
    Alvarez-Hamelin, J.I., Dall’Asta, L., Barrat, A., Vespignani, A.: k-core decomposition: a tool for the analysis of large scale internet graphs (2005)Google Scholar
  9. 9.
    Alvarez-Hamelin, J.I., Dall’Asta, L., Barrat, A., Vespignani, A.: \(k\)-core decomposition of internet graphs: Hierarchies, self-similarity and measurement biases. NHM 3(2), 371 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: WAW, pp. 25–37 (2009)Google Scholar
  11. 11.
    Angluin, D., Chen, J.: Learning a hidden graph using o( logn) queries per edge. J. Comput. Syst. Sci. 74(4), 546–556 (2008)zbMATHCrossRefGoogle Scholar
  12. 12.
    Aridhi, S., Brugnara, M., Montresor, A., Velegrakis, Y.: Distributed k-core decomposition and maintenance in large dynamic graphs. In: Proceedings of the 10th ACM International Conference on Distributed and Event-based Systems, DEBS’16, New York, NY, pp. 161–168. ACM (2016)Google Scholar
  13. 13.
    Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, Berlin (2008)zbMATHGoogle Scholar
  14. 14.
    Bao, J., Zheng, Y., Wilkie, D., Mokbel, M.: Recommendations in location-based social networks: a survey. GeoInformatica 19(3), 525–565 (2015)CrossRefGoogle Scholar
  15. 15.
    Bastian, M., Heymann, S., Jacomy, M., et al.: Gephi: an open source software for exploring and manipulating networks. ICWSM 8(2009), 361–362 (2009)Google Scholar
  16. 16.
    Batagelj, V., Mrvar, A., Zaveršnik, M.: Partitioning approach to visualization of large graphs. In: International Symposium on Graph Drawing, pp. 90–97. Springer (1999)Google Scholar
  17. 17.
    Batagelj, V., Zaversnik, M.: Generalized cores. CoRR, cs.DS/0202039 (2002)Google Scholar
  18. 18.
    Batagelj, V., Zaversnik, M.: An o(m) algorithm for cores decomposition of networks (2003). arXiv:cs/0310049
  19. 19.
    Benson, A.R., Gleich, D.F., Leskovec, J.: Higher-order organization of complex networks. Science 353(6295), 163–166 (2016)CrossRefGoogle Scholar
  20. 20.
    Bhawalkar, K., Kleinberg, J., Lewi, K., Roughgarden, T., Sharma, A.: Preventing unraveling in social networks: the anchored \(k\)-core problem. In: ICALP’11: Proceedings of the 39th International Colloquium Conference on Automata, Languages, and Programming, pp. 440–451 (2011)CrossRefGoogle Scholar
  21. 21.
    Bola, M., Sabel, B.: Dynamic reorganization of brain functional networks during cognition. NeuroImage 114, 03 (2015)CrossRefGoogle Scholar
  22. 22.
    Boldi, P., Vigna, S.: The webgraph framework I: compression techniques. In: Proceedings of the 13th International Conference on World Wide Web, WWW’04, New York, NY, pp. 595–602. ACM (2004)Google Scholar
  23. 23.
    Bonchi, F., Gullo, F., Kaltenbrunner, A.: Core Decomposition of Massive, Information-Rich Graphs, pp. 1–11. Springer, New York (2017)Google Scholar
  24. 24.
    Bonchi, F., Gullo, F., Kaltenbrunner, A., Volkovich, Y.: Core decomposition of uncertain graphs. In: KDD, pp. 1316–1325 (2014)Google Scholar
  25. 25.
    Bonchi, F., Khan, A., Severini, L.: Distance-generalized core decomposition. In: Proceedings of the 2019 ACM SIGMOD International Conference on Management of Data (2019)Google Scholar
  26. 26.
    Borzsony, S., Kossmann, D., Stocker, K.: The skyline operator. In: Proceedings 17th International Conference on Data Engineering, pp. 421–430 (2001)Google Scholar
  27. 27.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. In: Proceedings of the Seventh International Conference on World Wide Web 7, WWW7, pp. 107–117. Elsevier Science Publishers B. V., Amsterdam (1998)CrossRefGoogle Scholar
  28. 28.
    Brown, P., Feng, J.: Measuring user influence on twitter using modified k-shell decomposition. In: The Social Mobile Web, Volume WS-11-02 of AAAI Workshops. AAAI (2011)Google Scholar
  29. 29.
    Carmi, S., Havlin, S., Kirkpatrick, S., Shavitt, Y., Shir, E.: A model of internet topology using \(k\)-shell decomposition. PNAS 104(27), 11150–11154 (2007)CrossRefGoogle Scholar
  30. 30.
    Chang, L., Qin, L.: Cohesive Subgraph Computation over Large Sparse Graphs. Springer, Berlin (2018)zbMATHCrossRefGoogle Scholar
  31. 31.
    Chang, Q.-L.: Lijun: Minimum Degree-Based Core Decomposition. Springer Series in the Data Sciences, pp. 21–39. Springer, Berlin (2018)Google Scholar
  32. 32.
    Cheng, J., Ke, Y., Chu, S., Ozsu, M.T.: Efficient core decomposition in massive networks. In: ICDE, pp. 51–62 (2011)Google Scholar
  33. 33.
    Cheng, S.-T., Chen, Y.-C., Tsai, M.-S.: Using k-core decomposition to find cluster centers for k-means algorithm in graphx on spark. In: Proceedings of the 8-th International Conference on Cloud Computing, GRIDs, and Virtualization, pp. 93–98 (2017)Google Scholar
  34. 34.
    Cohen, J.: Trusses: cohesive subgraphs for social network analysis. National Security Agency Technical Report (2008)Google Scholar
  35. 35.
    Colomer-de Simón, P., Serrano, M.A., Beiró, M.G., Alvarez-Hamelin, J.I., Boguná, M.: Deciphering the global organization of clustering in real complex networks. Sci. Rep. 3, 2517 (2013)CrossRefGoogle Scholar
  36. 36.
    Cook, D.J., Holder, L.B.: Mining Graph Data. Wiley, Hoboken (2006)zbMATHCrossRefGoogle Scholar
  37. 37.
    Cui, W., Xiao, Y., Wang, H., Wang, W.: Local search of communities in large graphs. In: Proceedings of the 2014 ACM SIGMOD International Conference on Management of Data, pp. 991–1002 (2014)Google Scholar
  38. 38.
    Danisch, M., Chan, T.-H.H., Sozio, M.: Large scale density-friendly graph decomposition via convex programming. In: Proceedings of the 26th International Conference on World Wide Web, WWW’17, pp. 233–242 (2017)Google Scholar
  39. 39.
    Dean, J., Ghemawat, S.: Mapreduce: simplified data processing on large clusters. In: Proceedings of the 6th Conference on Symposium on Operating Systems Design & Implementation—Volume 6, OSDI’04, pp. 10–10. USENIX Association, Berkeley, CA (2004)Google Scholar
  40. 40.
    Ding, D., Li, H., Huang, Z., Mamoulis, N.: Efficient fault-tolerant group recommendation using alpha-beta-core. In: Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, CIKM’17. New York, NY, pp. 2047–2050. ACM (2017)Google Scholar
  41. 41.
    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: k-core organization of complex networks. Phys. Rev. Lett. 96, 040601 (2006)CrossRefGoogle Scholar
  42. 42.
    Eidsaa, M.: Core decomposition analysis of weighted biological networks. Ph.D. thesis, NTNU (2016)Google Scholar
  43. 43.
    Eidsaa, M., Almaas, E.: \(s\)-core network decomposition: a generalization of \(k\)-core analysis to weighted networks. Phys. Rev. E 88, 062819 (2013)CrossRefGoogle Scholar
  44. 44.
    Emerson, A.I., Andrews, S., Ahmed, I., Azis, T.K., Malek, J.A.: K-core decomposition of a protein domain co-occurrence network reveals lower cancer mutation rates for interior cores. J. Clin. Bioinform. 5(1), 1 (2015)CrossRefGoogle Scholar
  45. 45.
    ErdÅs, P., Hajnal, A.: On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hung. 17(1–2), 61–99 (1966)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Fang, Y., Cheng, R., Li, X., Luo, S., Hu, J.: Effective community search over large spatial graphs. Proc. VLDB Endow. 10(6), 709–720 (2017)CrossRefGoogle Scholar
  47. 47.
    Fang, Y., Cheng, R., Luo, S., Hu, J.: Effective community search for large attributed graphs. Proc. VLDB Endow. 9(12), 1233–1244 (2016)CrossRefGoogle Scholar
  48. 48.
    Farach-Colton, M., Tsai, M.-T.: Computing the degeneracy of large graphs. In: Latin American Symposium on Theoretical Informatics, pp. 250–260. Springer (2014)Google Scholar
  49. 49.
    Filho, H.A., Machicao, J., Bruno, O.M.: A hierarchical model of metabolic machinery based on the kcore decomposition of plant metabolic networks. PLoS ONE 13(5), 1–15 (2018)CrossRefGoogle Scholar
  50. 50.
    Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 40(1), 35–41 (1977)CrossRefGoogle Scholar
  51. 51.
    Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29(1), 24–32 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Galimberti, E., Barrat, A., Bonchi, F., Cattuto, C., Gullo, F.: Mining (maximal) span-cores from temporal networks. In: Proceedings of the 27th ACM International Conference on Information and Knowledge Management, pp. 107–116. ACM (2018)Google Scholar
  53. 53.
    Galimberti, E., Bonchi, F., Gullo, F.: Core decomposition and densest subgraph in multilayer networks. In: Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, CIKM’17, New York, NY, pp. 1807–1816. ACM (2017)Google Scholar
  54. 54.
    Garas, A., Schweitzer, F., Havlin, S.: A \(k\)-shell decomposition method for weighted networks. New J. Phys. 14(8), 083030 (2012)CrossRefGoogle Scholar
  55. 55.
    Garcia, D., Mavrodiev, P., Schweitzer, F.: Social resilience in online communities: the autopsy of friendster. In: COSN’13: Proceedings of the First ACM Conference on Online Social Networks, pp. 39–50 (2013)Google Scholar
  56. 56.
    Garcia-Algarra, J., Pastor, J., Mouronte, M.L., Galeano, J.: A structural approach to disentangle the visualization of bipartite biological networks. Complexity 1–11(02), 2018 (2018)Google Scholar
  57. 57.
    Garcia-Algarra, J., Pastor, J.M.M., Mouronte, M.L., Galeano, J.: Bipartgraph: an interactive application to plot bipartite ecological networks. bioRxiv (2017)Google Scholar
  58. 58.
    García-Algarra, J., Pastor, J., Iriondo, J., Galeano, J.: Ranking of critical species to preserve the functionality of mutualistic networks using the \(k\)-core decomposition. PeerJ 5, 3321 (2017)CrossRefGoogle Scholar
  59. 59.
    Giatsidis, C., Berberich, K., Thilikos, D.M., Vazirgiannis, M.: Visual exploration of collaboration networks based on graph degeneracy. In: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 1512–1515. ACM (2012)Google Scholar
  60. 60.
    Giatsidis, C., Cautis, B., Maniu, S., Thilikos, D.M., Vazirgiannis, M.: Quantifying trust dynamics in signed graphs, the s-cores approach. In: Proceedings of the 2014 SIAM International Conference on Data Mining, Philadelphia, Pennsylvania, USA, April 24–26, 2014, pp. 668–676 (2014)Google Scholar
  61. 61.
    Giatsidis, C., Malliaros, F.D., Thilikos, D.M., Vazirgiannis, M.: Corecluster: A degeneracy based graph clustering framework. In: AAAI’14: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 44–50 (2014)Google Scholar
  62. 62.
    Giatsidis, C., Thilikos, D.M., Vazirgiannis, M.: D-cores: measuring collaboration of directed graphs based on degeneracy. In: ICDM’11: Proceedings of the 11th IEEE International Conference on Data Mining, pp. 201–210 (2011)Google Scholar
  63. 63.
    Giatsidis, C., Thilikos, D.M., Vazirgiannis, M.: Evaluating cooperation in communities with the \(k\)-core structure. In: ASONAM’11: Proceedings of the International Conference on Advances in Social Networks Analysis and Mining, pp. 87–93 (2011)Google Scholar
  64. 64.
    Giatsidis, C., Thilikos, D.M., Vazirgiannis, M.: D-cores: measuring collaboration of directed graphs based on degeneracy. Knowl. Inf. Syst. 35(2), 311–343 (2013)CrossRefGoogle Scholar
  65. 65.
    Govindan, P., Soundarajan, S., Eliassi-Rad, T., Faloutsos, C.: Nimblecore: A space-efficient external memory algorithm for estimating core numbers. In: ASONAM, pp. 207–214. IEEE Computer Society (2016)Google Scholar
  66. 66.
    Govindan, P., Wang, C., Xu, C., Duan, H., Soundarajan, S.: The \(k\)-peak decomposition: mapping the global structure of graphs. In: Proceedings of the 26th International Conference on World Wide Web, WWW’17, pp. 1441–1450 (2017)Google Scholar
  67. 67.
    Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C.J., Sporns, O.: Mapping the structural core of human cerebral cortex. PLoS Biol. 6(7), e159 (2008)CrossRefGoogle Scholar
  68. 68.
    He, X., Zhao, H., Cai, W., Li, G.-G., Pei, F.-D.: Analyzing the structure of earthquake network by \(k\)-core decomposition. Phys. A 421, 34–43 (2015)CrossRefGoogle Scholar
  69. 69.
    Healy, J., Janssen, J., Milios, E., Aiello, W.: Characterization of graphs using degree cores. In: WAW’08: Algorithms and Models for the Web-Graph, pp. 137–148 (2008)Google Scholar
  70. 70.
    Hébert-Dufresne, L., Allard, A., Young, J.-G., Dubé, L.J.: Percolation on random networks with arbitrary \(k\)-core structure. Phys. Rev. E 88(6), 062820 (2013)CrossRefGoogle Scholar
  71. 71.
    Hu, X., Liu, F., Srinivasan, V., Thomo, A.: \(k\)-core decomposition on giraph and GraphChi. In: Barolli, L., Woungang, I., Hussain, O.K. (eds.) Advances in Intelligent Networking and Collaborative Systems, pp. 274–284. Springer, Cham (2018)CrossRefGoogle Scholar
  72. 72.
    Huang, X., Lu, W., Lakshmanan, L.V.: Truss decomposition of probabilistic graphs: semantics and algorithms. In: Proceedings of the 2016 ACM SIGMOD International Conference on Management of Data, SIGMOD’16, pp. 77–90 (2016)Google Scholar
  73. 73.
    Isaac, A.E., Sinha, S.: Analysis of core-periphery organization in protein contact networks reveals groups of structurally and functionally critical residues. J. Biosci. 40(4), 683–699 (2015)CrossRefGoogle Scholar
  74. 74.
    Kabir, H., Madduri, K.: Parallel \(k\)-core decomposition on multicore platforms. In: IPDPS Workshops, pp. 1482–1491. IEEE Computer Society (2017)Google Scholar
  75. 75.
    Kassiano, V., Gounaris, A., Papadopoulos, A.N., Tsichlas, K.: Mining uncertain graphs: an overview. In: Sellis, T., Oikonomou, K. (eds.) Algorithmic Aspects of Cloud Computing, pp. 87–116. Springer, Cham (2017)CrossRefGoogle Scholar
  76. 76.
    Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. In: Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining, KDD’03, pp. 137–146 (2003)Google Scholar
  77. 77.
    Khaouid, W., Barsky, M., Venkatesh, S., Thomo, A.: K-core decomposition of large networks on a single PC. PVLDB 9(1), 13–23 (2015)Google Scholar
  78. 78.
    Kirousis, L.M., Thilikos, D.M.: The linkage of a graph. SIAM J. Comput. 25(3), 626–647 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Kitsak, M., Gallos, L.K., Havlin, S., Liljerosand, F., Muchnik, L., Stanley, H.E., Makse, H.A.: Identification of influential spreaders in complex networks. Nat. Phys. 6, 888 (2010)CrossRefGoogle Scholar
  80. 80.
    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: The web as a graph. In: PODS (2000)Google Scholar
  81. 81.
    Kunegis, J., Lommatzsch, A., Bauckhage, C.: The slashdot zoo: mining a social network with negative edges. In: Proceedings of the 18th International Conference on World Wide Web, WWW’09, New York, NY, USA, pp. 741–750. ACM (2009)Google Scholar
  82. 82.
    Kunegis, J., Schmidt, S., Lommatzsch, A., Lerner, J., Luca, E.W.D., Albayrak, S.: Spectral analysis of signed graphs for clustering, prediction and visualization. In: SDM, pp. 559–570. SIAM (2010)Google Scholar
  83. 83.
    Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. J. ACM 22(4), 469–476 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Kyrola, A., Blelloch, G., Guestrin, C.: Graphchi: Large-scale graph computation on just a pc. In: Proceedings of the 10th USENIX Conference on Operating Systems Design and Implementation, OSDI’12, Berkeley, CA, USA, pp. 31–46. USENIX Association (2012)Google Scholar
  85. 85.
    Lahav, N., Ksherim, B., Ben-Simon, E., Maron-Katz, A., Cohen, R., Havlin, S.: K -shell decomposition reveals hierarchical cortical organization of the human brain. New J. Phys. 18(8), 083013 (2016)CrossRefGoogle Scholar
  86. 86.
    Laishram, R., Sariyüce, A.E., Eliassi-Rad, T., Pinar, A., Soundarajan, S.: Measuring and improving the core resilience of networks. In: Proceedings of the 2018 World Wide Web Conference, WWW’18, pp. 609–618 (2018)Google Scholar
  87. 87.
    Leskovec, J., Horvitz, E.: Planetary-scale views on a large instant-messaging network. In: WWW’08: Proceedings of the 17th International Conference on World Wide Web, pp. 915–924 (2008)Google Scholar
  88. 88.
    Leskovec, J., Huttenlocher, D., Kleinberg, J.: Signed networks in social media. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI’10, New York, NY, USA, pp. 1361–1370. ACM (2010)Google Scholar
  89. 89.
    Leskovec, J., Krevl, A.: SNAP datasets: Stanford large network dataset collection. http://snap.stanford.edu/data (2014)
  90. 90.
    Li, R., Su, J., Qin, L., Yu, J.X., Dai, Q.: Persistent community search in temporal networks. In: 2018 IEEE 34th International Conference on Data Engineering (ICDE), pp. 797–808 (2018)Google Scholar
  91. 91.
    Li, R.-H., Qin, L., Ye, F., Yu, J.X., Xiao, X., Xiao, N., Zheng, Z.: Skyline community search in multi-valued networks. In: Proceedings of the 2018 ACM SIGMOD International Conference on Management of Data, SIGMOD’18, New York, NY, USA, pp. 457–472. ACM (2018)Google Scholar
  92. 92.
    Li, R.-H., Qin, L., Yu, J.X., Mao, R.: Influential community search in large networks. Proc. VLDB Endow. 8(5), 509–520 (2015)CrossRefGoogle Scholar
  93. 93.
    Li, R.-H., Yu, J.X., Mao, R.: Efficient core maintenance in large dynamic graphs. IEEE Trans. Knowl. Data Eng. 26(10), 2453–2465 (2014)CrossRefGoogle Scholar
  94. 94.
    Lick, D.R., White, A.T.: \(k\)-degenerate graphs. Can. J. Math. 22, 1082–1096 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Lin, J.-H., Guo, Q., Dong, W.-Z., Tang, L.-Y., Liu, J.-G.: Identifying the node spreading influence with largest \(k\)-core values. Phys. Lett. A 378(45), 3279–3284 (2014)zbMATHCrossRefGoogle Scholar
  96. 96.
    Litvak, M., Last, M.: Graph-based keyword extraction for single-document summarization. In: Proceedings of the Workshop on Multi-source Multilingual Information Extraction and Summarization, pp. 17–24. Association for Computational Linguistics (2008)Google Scholar
  97. 97.
    Lü, L., Chen, D., Ren, X.-L., Zhang, Q.-M., Zhang, Y.-C., Zhou, T.: Vital nodes identification in complex networks. Phys. Rep. 650, 1–63 (2016)MathSciNetCrossRefGoogle Scholar
  98. 98.
    Lü, L., Zhou, T., Zhang, Q.-M., Stanley, H.E.: The \(h\)-index of a network node and its relation to degree and coreness. Nat. Commun. 7, 10168 (2016)CrossRefGoogle Scholar
  99. 99.
    Luo, F., Li, B., Wan, X.-F., Scheuermann, R.H.: Core and periphery structures in protein interaction networks. BMC Bioinform. 10(Suppl 4), s8 (2009)CrossRefGoogle Scholar
  100. 100.
    Malewicz, G., Austern, M.H., Bik, A.J., Dehnert, J.C., Horn, I., Leiser, N., Czajkowski, G.: Pregel: A system for large-scale graph processing. In: Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data, SIGMOD’10, New York, NY, USA, pp. 135–146. ACM (2010)Google Scholar
  101. 101.
    Malliaros, F.D., Papadopoulos, A.N., Vazirgiannis, M.: Core decomposition in graphs: concepts, algorithms and applications. In: EDBT. OpenProceedings.org, pp. 720–721 (2016)Google Scholar
  102. 102.
    Malliaros, F.D., Rossi, M.-E.G., Vazirgiannis, M.: Locating influential nodes in complex networks. Sci. Rep. 6, 19307 (2016)CrossRefGoogle Scholar
  103. 103.
    Malliaros, F.D., Vazirgiannis, M.: To stay or not to stay: modeling engagement dynamics in social graphs. In: 22nd ACM International Conference on Information and Knowledge Management, CIKM’13, pp. 469–478 (2013)Google Scholar
  104. 104.
    Malliaros, F.D., Vazirgiannis, M.: Vulnerability assessment in social networks under cascade-based node departures. EPL (Eur. Lett.) 110(6), 68006 (2015)CrossRefGoogle Scholar
  105. 105.
    Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30(3), 417–427 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Meladianos, P., Nikolentzos, G., Rousseau, F., Stavrakas, Y., Vazirgiannis, M.: Degeneracy-based real-time sub-event detection in twitter stream. In: ICWSM, pp. 248–257 (2015)Google Scholar
  107. 107.
    Meladianos, P., Tixier, A., Nikolentzos, I., Vazirgiannis, M.: Real-time keyword extraction from conversations. In: Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers, pp. 462–467 (2017)Google Scholar
  108. 108.
    Mihalcea, R., Tarau, P.: Textrank: bringing order into text. In: Proceedings of the 2004 Conference on Empirical Methods in Natural Language Processing (2004)Google Scholar
  109. 109.
    Montresor, A., De Pellegrini, F., Miorandi, D.: Distributed k-core decomposition. In: PODC, pp. 207–208 (2011)Google Scholar
  110. 110.
    Montresor, A., De Pellegrini, F., Miorandi, D.: Distributed \(k\)-core decomposition. IEEE Trans. Parallel Distrib. Syst. 24(2), 288–300 (2013)CrossRefGoogle Scholar
  111. 111.
    Morone, F., Burleson-Lesser, K., Vinutha, H., Sastry, S., Makse, H.A.: The jamming transition is a \(k\)-core percolation transition. Phys. A 516, 172–177 (2019)CrossRefGoogle Scholar
  112. 112.
    Morone, F., Ferraro, G., Makse, H.A.: The \(k\)-core as a predictor of structural collapse in mutualistic ecosystems. Nat. Phys. 10, 95–102 (2018)Google Scholar
  113. 113.
    Nikolentzos, G., Meladianos, P., Limnios, S., Vazirgiannis, M.: A degeneracy framework for graph similarity. In: IJCAI, pp. 2595–2601 (2018)Google Scholar
  114. 114.
    O’Brien, M.P., Sullivan, B.D.: Locally estimating core numbers. In: ICDM, pp. 460–469 (2014)Google Scholar
  115. 115.
    Parchas, P., Gullo, F., Papadias, D., Bonchi, F.: The pursuit of a good possible world: extracting representative instances of uncertain graphs. In: Proceedings of the 2014 ACM SIGMOD International Conference on Management of Data, pp. 967–978 (2014)Google Scholar
  116. 116.
    Parchas, P., Gullo, F., Papadias, D., Bonchi, F.: Uncertain graph processing through representative instances. ACM Trans. Database Syst. 40(3), 20:1–20:39 (2015)MathSciNetCrossRefGoogle Scholar
  117. 117.
    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)CrossRefGoogle Scholar
  118. 118.
    Pei, S., Makse, H.A.: Spreading dynamics in complex networks. J. Stat. Mech. Theory Exp. 2013(12), P12002 (2013)CrossRefGoogle Scholar
  119. 119.
    Pei, S., Muchnik, L., Andrade Jr., J.S., Zheng, Z., Makse, H.A.: Searching for superspreaders of information in real-world social media. Sci. Rep. 4, 5547 (2014)CrossRefGoogle Scholar
  120. 120.
    Pellegrini, M., Baglioni, M., Geraci, F.: Protein complex prediction for large protein protein interaction networks with the core & peel method. BMC Bioinform. 17(12), 372 (2016)CrossRefGoogle Scholar
  121. 121.
    Peng, Y., Zhang, Y., Zhang, W., Lin, X., Qin, L.: Efficient probabilistic \(k\)-core computation on uncertain graphs. In: 2018 IEEE 34th International Conference on Data Engineering (ICDE), pp. 1192–1203 (2018)Google Scholar
  122. 122.
    Phizicky, E.M., Fields, S.: Protein-protein interactions: methods for detection and analysis. Microbiol. Rev. 59(1), 94–123 (1995)Google Scholar
  123. 123.
    Potamias, M., Bonchi, F., Gionis, A., Kollios, G.: K-nearest neighbors in uncertain graphs. In: Proceedings of the VLDB Endowment, pp. 997–1008 (2010)CrossRefGoogle Scholar
  124. 124.
    Rousseau, F., Vazirgiannis, M.: Main core retention on graph-of-words for single-document keyword extraction. In: ECIR’15: Proceedings of the 37th European Conference on Information Retrieval, pp. 382–393 (2015)Google Scholar
  125. 125.
    Samu, D., Seth, A.K., Nowotny, T.: Influence of wiring cost on the large-scale architecture of human cortical connectivity. PLOS Comput. Biol. 10(4), 1–24 (2014)CrossRefGoogle Scholar
  126. 126.
    Sarıyüce, A.E., Gedik, B., Jacques-Silva, G., Wu, K.-L., Çatalyürek, Ü.V.: Incremental \(k\)-core decomposition: algorithms and evaluation. VLDB J. 25(3), 425–447 (2016)CrossRefGoogle Scholar
  127. 127.
    Saríyüce, A.E., Gedik, B., Jacques-Silva, G., Wu, K.-L., Çatalyürek, U.V.: Streaming algorithms for \(k\)-core decomposition. Proc. VLDB Endow. 6(6), 433–444 (2013)CrossRefGoogle Scholar
  128. 128.
    Sariyüce, A.E., Pinar, A.: Peeling bipartite networks for dense subgraph discovery. In: Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining, WSDM, pp. 504–512 (2018)Google Scholar
  129. 129.
    Sariyüce, A.E., Seshadhri, C., Pinar, A.: Local algorithms for hierarchical dense subgraph discovery. Proc. VLDB Endow. 12(1), 43–56 (2018)CrossRefGoogle Scholar
  130. 130.
    Sariyuce, A.E., Seshadhri, C., Pinar, A., Catalyurek, U.V.: Finding the hierarchy of dense subgraphs using nucleus decompositions. In: Proceedings of the 24th International Conference on World Wide Web, WWW’15, pp. 927–937 (2015)Google Scholar
  131. 131.
    Sarkar, S., Bhagwat, A., Mukherjee, A.: Core2vec: a core-preserving feature learning framework for networks. In: IEEE/ACM 2018 International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2018, pp. 487–490 (2018)Google Scholar
  132. 132.
    Schmidt, C., Pfister, H.D., Zdeborová, L.: Minimal sets to destroy the k-core in random networks. Phys. Rev. E 99(2), 022310 (2019)CrossRefGoogle Scholar
  133. 133.
    Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5, 269–287 (1983)MathSciNetCrossRefGoogle Scholar
  134. 134.
    Shailaja Dasari, N., Ranjan, D., Zubair, M.: Park: An efficient algorithm for k-core decomposition on multicore processors. Proceedings—2014 IEEE International Conference on Big Data, IEEE Big Data 2014, pp. 9–16 (2015)Google Scholar
  135. 135.
    Shanahan, M., Bingman, V., Shimizu, T., Wild, M., Güntürkün, O.: Large-scale network organization in the avian forebrain: a connectivity matrix and theoretical analysis. Front. Comput. Neurosci. 7, 89 (2013)CrossRefGoogle Scholar
  136. 136.
    Shin, K., Eliassi-Rad, T., Faloutsos, C.: Corescope: Graph mining using k-core analysis—patterns, anomalies and algorithms. In: ICDM, pp. 469–478. IEEE (2016)Google Scholar
  137. 137.
    Shin, K., Eliassi-Rad, T., Faloutsos, C.: Patterns and anomalies in k-cores of real-world graphs with applications. Knowl. Inf. Syst. 54(3), 677–710 (2018)CrossRefGoogle Scholar
  138. 138.
    Sozio, M., Gionis, A.: The community-search problem and how to plan a successful cocktail party. In: Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining—KDD’10, pp. 939–948. Exported from https://app.dimensions.ai on 27 April 2019 (2010)
  139. 139.
    Strouthopoulos, P., Papadopoulos, A.N.: Core discovery in hidden graphs. CoRR (to appear in Data and Knowledge Engineering). arXiv:1712.02827 (2017)
  140. 140.
    Tao, Y., Sheng, C., Li, J.: Finding maximum degrees in hidden bipartite graphs. In: Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data, SIGMOD’10, New York, NY, USA, pp. 891–902. ACM (2010)Google Scholar
  141. 141.
    Tatti, N., Gionis, A.: Density-friendly graph decomposition. In: WWW, pp. 1089–1099 (2015)Google Scholar
  142. 142.
    Tenenbaum, J.B., De Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  143. 143.
    Tixier, A., Malliaros, F.D., Vazirgiannis, M.: A graph degeneracy-based approach to keyword extraction. In: Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pp. 1860–1870. Association for Computational Linguistics (2016)Google Scholar
  144. 144.
    Tixier, A., Skianis, K., Vazirgiannis, M.: Gowvis: a web application for graph-of-words-based text visualization and summarization. In: Proceedings of ACL-2016 System Demonstrations, pp. 151–156 (2016)Google Scholar
  145. 145.
    Tsourakakis, C.E., Kang, U., Miller, G.L., Faloutsos, C.: Doulion: counting triangles in massive graphs with a coin. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 837–846. ACM (2009)Google Scholar
  146. 146.
    Ugander, J., Karrer, B., Backstrom, L., Marlow, C.: The anatomy of the facebook social graph. arXiv:1111.4503. Comment: 17 pp., 9 figures, 1 table (2011)
  147. 147.
    van den Heuvel, M.P., Sporns, O.: Rich-club organization of the human connectome. J. Neurosci. 31(44), 15775–15786 (2011)CrossRefGoogle Scholar
  148. 148.
    Verma, T., Russmann, F., Araújo, N., Nagler, J., Herrmann, H.: Emergence of core-peripheries in networks. Nat. Commun. 7, 10441 (2016)CrossRefGoogle Scholar
  149. 149.
    Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar
  150. 150.
    Wang, J., Cheng, J.: Truss decomposition in massive networks. Proc. VLDB Endow. 5(9), 812–823 (2012)CrossRefGoogle Scholar
  151. 151.
    Wang, K., Cao, X., Lin, X., Zhang, W., Qin, L.: Efficient computing of radius-bounded k-cores. In: 2018 IEEE 34th International Conference on Data Engineering (ICDE), pp. 233–244 (2018)Google Scholar
  152. 152.
    Wang, N., Yu, D., Jin, H., Qian, C., Xie, X., Hua, Q.: Parallel algorithm for core maintenance in dynamic graphs. In: 2017 IEEE 37th International Conference on Distributed Computing Systems (ICDCS), vol. 00, pp. 2366–2371 (2017)Google Scholar
  153. 153.
    Wen, D., Qin, L., Zhang, Y., Lin, X., Yu, J.X.: I/O efficient core graph decomposition at web scale. In: 2016 IEEE 32nd International Conference on Data Engineering (ICDE), pp. 133–144. IEEE (2016)Google Scholar
  154. 154.
    Wen, D., Qin, L., Zhang, Y., Lin, X., Yu, J.X.: I/o efficient core graph decomposition: application to degeneracy ordering. IEEE Trans. Knowl. Data Eng. 31(1), 75–90 (2019)CrossRefGoogle Scholar
  155. 155.
    White, T.: Hadoop: The Definitive Guide, 4th edn. O’Reilly, Sebastopol (2015)Google Scholar
  156. 156.
    Wood, C.I., Hicks, I.V.: The minimal k-core problem for modeling k-assemblies. J. Math. Neurosci. (JMN) 5(1), 14 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  157. 157.
    Wu, H., Cheng, J., Lu, Y., Ke, Y., Huang, Y., Yan, D., Wu, H.: Core decomposition in large temporal graphs. In: BigData, pp. 649–658. IEEE (2015)Google Scholar
  158. 158.
    Yan, D., Cheng, J., Lu, Y., Ng, W.: Blogel: a block-centric framework for distributed computation on real-world graphs. Proc. VLDB Endow. 7(14), 1981–1992 (2014)CrossRefGoogle Scholar
  159. 159.
    Yiu, M.L., Lo, E., Wang, J.: Identifying the most connected vertices in hidden bipartite graphs using group testing. IEEE Trans. Knowl. Data Eng. 25, 2245–2256 (2013)CrossRefGoogle Scholar
  160. 160.
    Zaharia, M., Xin, R.S., Wendell, P., Das, T., Armbrust, M., Dave, A., Meng, X., Rosen, J., Venkataraman, S., Franklin, M.J., Ghodsi, A., Gonzalez, J., Shenker, S., Stoica, I.: Apache spark: a unified engine for big data processing. Commun. ACM 59(11), 56–65 (2016)CrossRefGoogle Scholar
  161. 161.
    Zdeborová, L., Zhang, P., Zhou, H.-J.: Fast and simple decycling and dismantling of networks. Sci. Rep. 6, 37954 (2016)CrossRefGoogle Scholar
  162. 162.
    Zhang, F., Zhang, W., Zhang, Y., Qin, L., Lin, X.: Olak: an efficient algorithm to prevent unraveling in social networks. Proc. VLDB Endow. 10(6), 649–660 (2017)CrossRefGoogle Scholar
  163. 163.
    Zhang, F., Zhang, Y., Qin, L., Zhang, W., Lin, X.: Finding critical users for social network engagement: the collapsed k-core problem. In: Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, pp. 245–251 (2017)Google Scholar
  164. 164.
    Zhang, F., Zhang, Y., Qin, L., Zhang, W., Lin, X.: When engagement meets similarity: efficient (k, r)-core computation on social networks. Proc. VLDB Endow. 10(10), 998–1009 (2017)CrossRefGoogle Scholar
  165. 165.
    Zhang, G.-Q., Zhang, G.-Q., Yang, Q.-F., Cheng, S.-Q., Zhou, T.: Evolution of the Internet and its cores. New J. Phys. 10(12), 123027+ (2008)CrossRefGoogle Scholar
  166. 166.
    Zhang, Y., Parthasarathy, S.: Extracting analyzing and visualizing triangle k-core motifs within networks. In: ICDE’12: Proceedings of the 2012 IEEE 28th International Conference on Data Engineering, pp. 1049–1060 (2012)Google Scholar
  167. 167.
    Zhang, Y., Yu, J.X., Zhang, Y., Qin, L.: A fast order-based approach for core maintenance. In: 2017 IEEE 33rd International Conference on Data Engineering (ICDE), pp. 337–348 (2017)Google Scholar
  168. 168.
    Zhuo-Ming, R., Jian-Guo, L., Feng, S., Zhao-Long, H., Qiang, G.: Analysis of the spreading influence of the nodes with minimum k-shell value in complex networks. Acta Phys. Sin. 62(10), 108902 (2013)Google Scholar
  169. 169.
    Zlatić, V., Garlaschelli, D., Caldarelli, G.: Networks with arbitrary edge multiplicities. EPL (Europhys. Lett.) 97(2), 28005 (2012)CrossRefGoogle Scholar
  170. 170.
    Zou, Z., Zhu, R.: Truss decomposition of uncertain graphs. Knowl. Inf. Syst. 50(1), 197–230 (2017)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Visual Computing, CentraleSupélecUniversity of Paris-Saclay and Inria SaclayGif-Sur-YvetteFrance
  2. 2.Computer Science LaboratoryÉcole PolytechniquePalaiseauFrance
  3. 3.School of InformaticsAristotle University of ThessalonikiThessalonikiGreece
  4. 4.Department of InformaticsAthens University of Economics and BusinessAthensGreece

Personalised recommendations