Advertisement

The core decomposition of networks: theory, algorithms and applications

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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Notes

  1. 1.

    www.epinions.com.

  2. 2.

    www.wikipedia.org.

  3. 3.

    https://akka.io.

  4. 4.

    The telencephalon is the most highly developed and anterior part of the forebrain, composed mainly of the cerebral hemispheres (https://en.wikipedia.org/wiki/Cerebrum).

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)

  2. 2.

    Aggarwal, C.C. (ed.): Social Network Data Analytics. Springer, Berlin (2011)

  3. 3.

    Aggarwal, C.C., Wang, H.: Managing and Mining Graph Data. Springer, Berlin (2010)

  4. 4.

    Akbas, E., Zhao, P.: Truss-based community search: a truss-equivalence based indexing approach. Proc. VLDB Endow. 10(11), 1298–1309 (2017)

  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)

  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)

  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)

  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)

  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)

  10. 10.

    Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: WAW, pp. 25–37 (2009)

  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)

  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)

  13. 13.

    Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, Berlin (2008)

  14. 14.

    Bao, J., Zheng, Y., Wilkie, D., Mokbel, M.: Recommendations in location-based social networks: a survey. GeoInformatica 19(3), 525–565 (2015)

  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)

  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)

  17. 17.

    Batagelj, V., Zaversnik, M.: Generalized cores. CoRR, cs.DS/0202039 (2002)

  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)

  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)

  21. 21.

    Bola, M., Sabel, B.: Dynamic reorganization of brain functional networks during cognition. NeuroImage 114, 03 (2015)

  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)

  23. 23.

    Bonchi, F., Gullo, F., Kaltenbrunner, A.: Core Decomposition of Massive, Information-Rich Graphs, pp. 1–11. Springer, New York (2017)

  24. 24.

    Bonchi, F., Gullo, F., Kaltenbrunner, A., Volkovich, Y.: Core decomposition of uncertain graphs. In: KDD, pp. 1316–1325 (2014)

  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)

  26. 26.

    Borzsony, S., Kossmann, D., Stocker, K.: The skyline operator. In: Proceedings 17th International Conference on Data Engineering, pp. 421–430 (2001)

  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)

  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)

  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)

  30. 30.

    Chang, L., Qin, L.: Cohesive Subgraph Computation over Large Sparse Graphs. Springer, Berlin (2018)

  31. 31.

    Chang, Q.-L.: Lijun: Minimum Degree-Based Core Decomposition. Springer Series in the Data Sciences, pp. 21–39. Springer, Berlin (2018)

  32. 32.

    Cheng, J., Ke, Y., Chu, S., Ozsu, M.T.: Efficient core decomposition in massive networks. In: ICDE, pp. 51–62 (2011)

  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)

  34. 34.

    Cohen, J.: Trusses: cohesive subgraphs for social network analysis. National Security Agency Technical Report (2008)

  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)

  36. 36.

    Cook, D.J., Holder, L.B.: Mining Graph Data. Wiley, Hoboken (2006)

  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)

  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)

  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)

  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)

  41. 41.

    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: k-core organization of complex networks. Phys. Rev. Lett. 96, 040601 (2006)

  42. 42.

    Eidsaa, M.: Core decomposition analysis of weighted biological networks. Ph.D. thesis, NTNU (2016)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  50. 50.

    Freeman, L.C.: A set of measures of centrality based on betweenness. Sociometry 40(1), 35–41 (1977)

  51. 51.

    Freuder, E.C.: A sufficient condition for backtrack-free search. J. ACM 29(1), 24–32 (1982)

  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)

  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)

  54. 54.

    Garas, A., Schweitzer, F., Havlin, S.: A \(k\)-shell decomposition method for weighted networks. New J. Phys. 14(8), 083030 (2012)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  74. 74.

    Kabir, H., Madduri, K.: Parallel \(k\)-core decomposition on multicore platforms. In: IPDPS Workshops, pp. 1482–1491. IEEE Computer Society (2017)

  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)

  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)

  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)

  78. 78.

    Kirousis, L.M., Thilikos, D.M.: The linkage of a graph. SIAM J. Comput. 25(3), 626–647 (1996)

  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)

  80. 80.

    Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E.: The web as a graph. In: PODS (2000)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  94. 94.

    Lick, D.R., White, A.T.: \(k\)-degenerate graphs. Can. J. Math. 22, 1082–1096 (1970)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  102. 102.

    Malliaros, F.D., Rossi, M.-E.G., Vazirgiannis, M.: Locating influential nodes in complex networks. Sci. Rep. 6, 19307 (2016)

  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)

  104. 104.

    Malliaros, F.D., Vazirgiannis, M.: Vulnerability assessment in social networks under cascade-based node departures. EPL (Eur. Lett.) 110(6), 68006 (2015)

  105. 105.

    Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30(3), 417–427 (1983)

  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)

  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)

  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)

  109. 109.

    Montresor, A., De Pellegrini, F., Miorandi, D.: Distributed k-core decomposition. In: PODC, pp. 207–208 (2011)

  110. 110.

    Montresor, A., De Pellegrini, F., Miorandi, D.: Distributed \(k\)-core decomposition. IEEE Trans. Parallel Distrib. Syst. 24(2), 288–300 (2013)

  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)

  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)

  113. 113.

    Nikolentzos, G., Meladianos, P., Limnios, S., Vazirgiannis, M.: A degeneracy framework for graph similarity. In: IJCAI, pp. 2595–2601 (2018)

  114. 114.

    O’Brien, M.P., Sullivan, B.D.: Locally estimating core numbers. In: ICDM, pp. 460–469 (2014)

  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)

  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)

  117. 117.

    Pastor-Satorras, R., Vespignani, A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)

  118. 118.

    Pei, S., Makse, H.A.: Spreading dynamics in complex networks. J. Stat. Mech. Theory Exp. 2013(12), P12002 (2013)

  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)

  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)

  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)

  122. 122.

    Phizicky, E.M., Fields, S.: Protein-protein interactions: methods for detection and analysis. Microbiol. Rev. 59(1), 94–123 (1995)

  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)

  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)

  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)

  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)

  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)

  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)

  129. 129.

    Sariyüce, A.E., Seshadhri, C., Pinar, A.: Local algorithms for hierarchical dense subgraph discovery. Proc. VLDB Endow. 12(1), 43–56 (2018)

  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)

  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)

  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)

  133. 133.

    Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5, 269–287 (1983)

  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)

  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)

  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)

  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)

  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)

  141. 141.

    Tatti, N., Gionis, A.: Density-friendly graph decomposition. In: WWW, pp. 1089–1099 (2015)

  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)

  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)

  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)

  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)

  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)

  148. 148.

    Verma, T., Russmann, F., Araújo, N., Nagler, J., Herrmann, H.: Emergence of core-peripheries in networks. Nat. Commun. 7, 10441 (2016)

  149. 149.

    Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)

  150. 150.

    Wang, J., Cheng, J.: Truss decomposition in massive networks. Proc. VLDB Endow. 5(9), 812–823 (2012)

  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)

  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)

  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)

  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)

  155. 155.

    White, T.: Hadoop: The Definitive Guide, 4th edn. O’Reilly, Sebastopol (2015)

  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)

  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)

  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)

  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)

  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)

  161. 161.

    Zdeborová, L., Zhang, P., Zhou, H.-J.: Fast and simple decycling and dismantling of networks. Sci. Rep. 6, 37954 (2016)

  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)

  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)

  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)

  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)

  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)

  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)

  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)

  169. 169.

    Zlatić, V., Garlaschelli, D., Caldarelli, G.: Networks with arbitrary edge multiplicities. EPL (Europhys. Lett.) 97(2), 28005 (2012)

  170. 170.

    Zou, Z., Zhu, R.: Truss decomposition of uncertain graphs. Knowl. Inf. Syst. 50(1), 197–230 (2017)

Download references

Author information

Correspondence to Apostolos N. Papadopoulos.

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

Malliaros, F.D., Giatsidis, C., Papadopoulos, A.N. et al. The core decomposition of networks: theory, algorithms and applications. The VLDB Journal (2019). https://doi.org/10.1007/s00778-019-00587-4

Download citation

Keywords

  • Core decomposition
  • Graph mining
  • Graph degeneracy
  • Graph theory
  • Algorithms