{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T22:26:07Z","timestamp":1790115967726,"version":"4.0.1"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T00:00:00Z","timestamp":1676592000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100021171","name":"Basic and Applied Basic Research Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2019A1515110473"],"award-info":[{"award-number":["2019A1515110473"]}],"id":[{"id":"10.13039\/501100021171","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62002235"],"award-info":[{"award-number":["62002235"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2019B151530001"],"award-info":[{"award-number":["2019B151530001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61729201"],"award-info":[{"award-number":["61729201"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["The VLDB Journal"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Uncertain butterflies are one of, if not the, most important graphlet structures on uncertain bipartite networks. In this paper, we examine the uncertain butterfly structure (in which the existential probability of the graphlet is greater than or equal to a threshold parameter), as well as the global Uncertain Butterfly Counting Problem (to count the total number of these instances over an entire network). To solve this task, we propose a non-trivial exact baseline (\n                    <jats:italic>UBFC<\/jats:italic>\n                    ), as well as an improved algorithm (\n                    <jats:italic>IUBFC<\/jats:italic>\n                    ) which we show to be faster both theoretically and practically. We also design two sampling frameworks (\n                    <jats:italic>UBS<\/jats:italic>\n                    and\n                    <jats:italic>PES<\/jats:italic>\n                    ) which can sample either a vertex, edge or wedge from the network uniformly and estimate the global count quickly. Furthermore, a notable butterfly-based community structure which has been examined in the past is the\n                    <jats:italic>k<\/jats:italic>\n                    -bitruss. We adapt this community structure onto the uncertain bipartite graph setting and introduce the Uncertain Bitruss Decomposition Problem (which can be used to directly answer any\n                    <jats:italic>k<\/jats:italic>\n                    -bitruss search query for any\n                    <jats:italic>k<\/jats:italic>\n                    ). We then propose an exact algorithm (\n                    <jats:italic>UBitD<\/jats:italic>\n                    ) to solve our problem with three variations in deriving the initial uncertain support. Using a range of networks with different edge existential probability distributions, we validate the efficiency and effectiveness of our solutions.\n                  <\/jats:p>","DOI":"10.1007\/s00778-023-00782-4","type":"journal-article","created":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T13:38:42Z","timestamp":1676641122000},"page":"1013-1036","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Butterfly counting and bitruss decomposition on uncertain bipartite graphs"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7063-2017","authenticated-orcid":false,"given":"Alexander","family":"Zhou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yue","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lei","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,2,17]]},"reference":[{"key":"782_CR1","doi-asserted-by":"publisher","unstructured":"Abiteboul, S., Kanellakis, P., Grahne, G.: On the representation and querying of sets of possible worlds. SIGMOD pp. 34\u201348 (1987). https:\/\/doi.org\/10.1145\/38713.38724","DOI":"10.1145\/38713.38724"},{"issue":"2","key":"782_CR2","doi-asserted-by":"publisher","DOI":"10.1002\/widm.1226","volume":"8","author":"M Al Hasan","year":"2018","unstructured":"Al Hasan, M., Dave, V.S.: Triangle counting in large networks: a review. WIREs DMKD 8(2), e1226 (2018). https:\/\/doi.org\/10.1002\/widm.1226","journal-title":"WIREs DMKD"},{"issue":"6","key":"782_CR3","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1101\/gr.2203804","volume":"14","author":"S Asthana","year":"2004","unstructured":"Asthana, S., King, O.D., Gibbons, F.D., Roth, F.P.: Predicting protein complex membership using probabilistic network reliability. Genome Res. 14(6), 1170\u20131175 (2004)","journal-title":"Genome Res."},{"issue":"1","key":"782_CR4","first-page":"43","volume":"19","author":"JE Bartlett","year":"2001","unstructured":"Bartlett, J.E., Kortlik, J.W., Higgins, C.C.: Organizational research: determining appropriate sample size in survey research. ITLPJ 19(1), 43\u201350 (2001)","journal-title":"ITLPJ"},{"key":"782_CR5","doi-asserted-by":"crossref","unstructured":"Becchetti, L., Boldi, P., Castillo, C., Gionis, A.: Efficient semi-streaming algorithms for local triangle counting in massive graphs. In: SIGKDD, pp. 16\u201324 (2008)","DOI":"10.1145\/1401890.1401898"},{"key":"782_CR6","doi-asserted-by":"crossref","unstructured":"Bonchi, F., Gullo, F., Kaltenbrunner, A., Volkovich, Y.: Core decomposition of uncertain graphs. In: SIGKDD, pp. 1316\u20131325 (2014)","DOI":"10.1145\/2623330.2623655"},{"issue":"10","key":"782_CR7","doi-asserted-by":"publisher","first-page":"1751","DOI":"10.14778\/3401960.3401971","volume":"13","author":"Y Che","year":"2020","unstructured":"Che, Y., Lai, Z., Sun, S., Wang, Y., Luo, Q.: Accelerating truss decomposition on heterogeneous processors. PVLDB 13(10), 1751\u20131764 (2020). https:\/\/doi.org\/10.14778\/3401960.3401971","journal-title":"PVLDB"},{"key":"782_CR8","doi-asserted-by":"publisher","unstructured":"Chen, P.L., Chou, C.K., Chen, M.S.: Distributed algorithms for k-truss decomposition. Big Data, pp. 471\u2013480 (2014). https:\/\/doi.org\/10.1109\/BigData.2014.7004264","DOI":"10.1109\/BigData.2014.7004264"},{"key":"782_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pcbi.1005557","volume":"13","author":"T Dallas","year":"2017","unstructured":"Dallas, T., Park, A.W., Drake, J.M.: Predicting cryptic links in host-parasite networks. PLoS Comput. Biol. 13, 1\u201315 (2017)","journal-title":"PLoS Comput. Biol."},{"issue":"3","key":"782_CR10","doi-asserted-by":"publisher","first-page":"596","DOI":"10.1145\/28869.28874","volume":"34","author":"ML Fredman","year":"1987","unstructured":"Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596\u2013615 (1987)","journal-title":"J. ACM"},{"issue":"6","key":"782_CR11","first-page":"667","volume":"12","author":"K Han","year":"2019","unstructured":"Han, K., Gui, F., Xiao, X., Tang, J., He, Y., Cao, Z., Huang, H.: Efficient and effective algorithms for clustering uncertain graphs. PVLDB 12(6), 667\u2013680 (2019)","journal-title":"PVLDB"},{"key":"782_CR12","doi-asserted-by":"crossref","unstructured":"Huang, X., Cheng, H., Qin, L., Tian, W., Yu, J.X.: Querying k-truss community in large and dynamic graphs. In: SIGMOD, pp. 1311\u20131322 (2014)","DOI":"10.1145\/2588555.2610495"},{"key":"782_CR13","doi-asserted-by":"publisher","unstructured":"Huang, X., Lu, W., Lakshmanan, L.: Truss decomposition of probabilistic graphs: Semantics and algorithms. In: SIGMOD, pp. 77\u201390 (2016). https:\/\/doi.org\/10.1145\/2882903.2882913","DOI":"10.1145\/2882903.2882913"},{"key":"782_CR14","doi-asserted-by":"publisher","unstructured":"Kabir, H., Madduri, K.: Shared-memory graph truss decomposition. In: HiPC, pp. 13\u201322 (2017). https:\/\/doi.org\/10.1109\/HiPC.2017.00012","DOI":"10.1109\/HiPC.2017.00012"},{"issue":"12","key":"782_CR15","first-page":"2042","volume":"8","author":"A Khan","year":"2015","unstructured":"Khan, A., Chen, L.: On uncertain graphs modelling and queries. PVLDB 8(12), 2042\u20132043 (2015)","journal-title":"PVLDB"},{"key":"782_CR16","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1007\/s13278-013-0122-z","volume":"3","author":"N Korovaiko","year":"2013","unstructured":"Korovaiko, N., Thomo, A.: Trust prediction from user-item ratings. Soc. Netw. Anal. Min. 3, 749\u2013759 (2013)","journal-title":"Soc. Netw. Anal. Min."},{"issue":"12","key":"782_CR17","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1371\/journal.pcbi.1009629","volume":"17","author":"JR Larsen","year":"2021","unstructured":"Larsen, J.R., Martin, M.R., Martin, J.D., Hicks, J.B., Kuhn, P.: Modeling the onset of symptoms of covid-19: effects of sars-cov-2 variant. PLOS Comput. Biol. 17(12), 89 (2021)","journal-title":"PLOS Comput. Biol."},{"key":"782_CR18","doi-asserted-by":"crossref","unstructured":"Li, R.H., Yu, J.X., Mao, R., Jin, T.: Efficient and accurate query evaluation on uncertain graphs via recursive stratified sampling. In: ICDE, pp. 892\u2013903 (2014)","DOI":"10.1109\/ICDE.2014.6816709"},{"key":"782_CR19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.72.056127","author":"P Lind","year":"2005","unstructured":"Lind, P., Gonzalez, M.C., Herrmann, H.: Cycles and clustering in bipartite networks. Phys. Rev. E. (2005). https:\/\/doi.org\/10.1103\/PhysRevE.72.056127","journal-title":"Phys. Rev. E."},{"issue":"5","key":"782_CR20","doi-asserted-by":"publisher","first-page":"1075","DOI":"10.1007\/s00778-020-00606-9","volume":"29","author":"B Liu","year":"2020","unstructured":"Liu, B., Yuan, L., Lin, X., Qin, L., Zhang, W., Zhou, J.: Efficient ($$\\alpha $$, $$\\beta $$)-core computation in bipartite graphs. VLDBJ 29(5), 1075\u20131099 (2020)","journal-title":"VLDBJ"},{"key":"782_CR21","doi-asserted-by":"publisher","first-page":"1075","DOI":"10.1007\/s00778-020-00606-9","volume":"29","author":"B Liu","year":"2020","unstructured":"Liu, B., Yuan, L., Lin, X., Qin, L., Zhang, W., Zhou, J.: Efficient $$(\\alpha, \\beta )$$-core computation in bipartite graphs. VLDBJ 29, 1075\u20131099 (2020)","journal-title":"VLDBJ"},{"issue":"9","key":"782_CR22","first-page":"1359","volume":"13","author":"B Lyu","year":"2020","unstructured":"Lyu, B., Qin, L., Lin, X., Zhang, Y., Qian, Z., Zhou, J.: Maximum biclique search at billion scale. PVLDB 13(9), 1359\u20131372 (2020)","journal-title":"PVLDB"},{"issue":"2","key":"782_CR23","first-page":"155","volume":"13","author":"C Ma","year":"2019","unstructured":"Ma, C., Cheng, R., Lakshamanan, L.V.S., Grubernmannm, T., Fang, Y., Li, X.: Linc: a motif counting algorithm for uncertain graphs. PVLDB 13(2), 155\u2013168 (2019)","journal-title":"PVLDB"},{"key":"782_CR24","doi-asserted-by":"crossref","unstructured":"Parchas, P., Gullo, F., Papadias, D., Bonchi, F.: The pursuit of a good possible world: Extracting representative instances of uncertain graphs. In: SIGMOD, pp. 967\u2013978 (2014)","DOI":"10.1145\/2588555.2593668"},{"issue":"4","key":"782_CR25","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1093\/gigascience\/giy014","volume":"7","author":"GA Pavlopoulos","year":"2018","unstructured":"Pavlopoulos, G.A., Kontou, P.I., Pavlopoulou, A., Bouyioukos, C., Markou, E., Bagos, P.G.: Bipartite graphs in systems biology and medicine: a survey of methods and applications. GigaScience 7(4), 87 (2018)","journal-title":"GigaScience"},{"issue":"3","key":"782_CR26","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1016\/s0166-218x(03)00333-0","volume":"131","author":"R Peeters","year":"2003","unstructured":"Peeters, R.: The maximum edge biclique problem is np-complete. Discret. Appl. Math. 131(3), 651\u2013654 (2003). https:\/\/doi.org\/10.1016\/s0166-218x(03)00333-0","journal-title":"Discret. Appl. Math."},{"issue":"1","key":"782_CR27","doi-asserted-by":"publisher","first-page":"23","DOI":"10.3390\/a12010023","volume":"12","author":"C Phillips","year":"2019","unstructured":"Phillips, C., Wang, K., Baker, E., Bubier, J., Chesler, E., Langston, M.: On finding and enumerating maximal and maximum k-partite cliques in k-partite graphs. Algorithms 12(1), 23 (2019). https:\/\/doi.org\/10.3390\/a12010023","journal-title":"Algorithms"},{"issue":"1","key":"782_CR28","first-page":"997","volume":"3","author":"M Potamias","year":"2010","unstructured":"Potamias, M., Bonchi, F., Gionis, A., Kollios, G.: k-nearest neighbors in uncertain graphs. PVLDB 3(1), 997\u20131008 (2010)","journal-title":"PVLDB"},{"key":"782_CR29","doi-asserted-by":"publisher","unstructured":"Rahman, M., Al\u00a0Hasan, M.: Sampling triples from restricted networks using mcmc strategy. In: CIKM, pp. 1519\u20131528 (2014). https:\/\/doi.org\/10.1145\/2661829.2662075","DOI":"10.1145\/2661829.2662075"},{"key":"782_CR30","first-page":"2150","volume":"24","author":"SV Sanei-Mehri","year":"2018","unstructured":"Sanei-Mehri, S.V., Sariyuce, A.E., Tirthapura, S.: Butterfly counting in bipartite networks. SIGKDD 24, 2150\u20132160 (2018)","journal-title":"SIGKDD"},{"key":"782_CR31","doi-asserted-by":"publisher","unstructured":"Sanei-Mehri, S.V., Zhang, Y., Sariy\u00fcce, A.E., Tirthapura, S.: Fleet: Butterfly estimation from a bipartite graph stream. In: CIKM, pp. 1201\u20131210 (2019). https:\/\/doi.org\/10.1145\/3357384.3357983","DOI":"10.1145\/3357384.3357983"},{"key":"782_CR32","unstructured":"Sarwar, B., Karypis, G., Konstan, J., Riedl, J.: Item-based collaborative filtering recommendation algorithms. WWW pp. 285\u2013295 (2010)"},{"issue":"2","key":"782_CR33","doi-asserted-by":"publisher","first-page":"265","DOI":"10.7155\/jgaa.00108","volume":"9","author":"T Schank","year":"2005","unstructured":"Schank, T., Wagner, D.: Approximating clustering-coefficient and transitivity. J. Gr. Algorithms Appl. 9(2), 265\u2013275 (2005)","journal-title":"J. Gr. Algorithms Appl."},{"key":"782_CR34","doi-asserted-by":"crossref","unstructured":"Shi, J., Shun, J.: Parallel algorithms for butterfly computations. In: APoCS, pp. 16\u201330 (2020)","DOI":"10.1137\/1.9781611976021.2"},{"key":"782_CR35","doi-asserted-by":"crossref","unstructured":"Smith, S., Liu, X., Ahmed, N.K., Tom, A.S., Petrini, F., Karypis, G.: Truss decomposition on shared-memory parallel systems. In: HPEC, pp. 1\u20136 (2017)","DOI":"10.1109\/HPEC.2017.8091049"},{"issue":"2","key":"782_CR36","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1287\/trsc.1090.0303","volume":"44","author":"I Sungur","year":"2010","unstructured":"Sungur, I., Ren, Y., Ord\u00f3\u00f1ez, F., Dessouky, M., Zhong, H.: A model and algorithm for the courier delivery problem with uncertainty. Transp. Sci. 44(2), 193\u2013205 (2010)","journal-title":"Transp. Sci."},{"key":"782_CR37","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1186\/1471-2105-7-360","volume":"7","author":"S Suthram","year":"2006","unstructured":"Suthram, S., Shlomi, T., Ruppin, E., Sharan, R., Ideker, T.: A direct comparison of protein interaction confidence assignment schemes. BMC Bioinf. 7, 360\u2013370 (2006)","journal-title":"BMC Bioinf."},{"issue":"9","key":"782_CR38","doi-asserted-by":"publisher","first-page":"972","DOI":"10.1109\/32.92917","volume":"17","author":"M Vose","year":"1991","unstructured":"Vose, M.: A linear algorithm for generating random numbers with a given distribution. IEEE Trans. Softw. Eng. 17(9), 972\u2013975 (1991). https:\/\/doi.org\/10.1109\/32.92917","journal-title":"IEEE Trans. Softw. Eng."},{"issue":"8","key":"782_CR39","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1049\/el:19740097","volume":"10","author":"AJ Walker","year":"1974","unstructured":"Walker, A.J.: New fast method for generating discrete random numbers with arbitrary frequency distributions. Electron. Lett. 10(8), 127\u2013128 (1974)","journal-title":"Electron. Lett."},{"key":"782_CR40","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1098\/rstb.2016.0095","volume":"372","author":"JG Walker","year":"2017","unstructured":"Walker, J.G., Plein, M., Morgan, E.R., Vesk, P.A.: Uncertain links in host-parasite networks: lessons for parasite transmission in a multi-host system. Philos. Trans. R. Soc. B 372, 86 (2017)","journal-title":"Philos. Trans. R. Soc. B"},{"issue":"9","key":"782_CR41","first-page":"812","volume":"5","author":"J Wang","year":"2012","unstructured":"Wang, J., Cheng, J.: Truss decomposition in massive networks. PVLDB 5(9), 812\u2013823 (2012)","journal-title":"PVLDB"},{"key":"782_CR42","doi-asserted-by":"publisher","unstructured":"Wang, J., Fu, A.W., Cheng, J.: Rectangle counting in large bipartite graphs. In: BigData, pp. 17\u201324 (2014). https:\/\/doi.org\/10.1109\/BigData.Congress.2014.13","DOI":"10.1109\/BigData.Congress.2014.13"},{"issue":"10","key":"782_CR43","doi-asserted-by":"publisher","first-page":"1139","DOI":"10.14778\/3339490.3339497","volume":"12","author":"K Wang","year":"2019","unstructured":"Wang, K., Lin, X., Qin, L., Zhang, W., Zhang, Y.: Vertex priority based butterfly counting for large-scale bipartite networks. PVLDB 12(10), 1139\u20131152 (2019). https:\/\/doi.org\/10.14778\/3339490.3339497","journal-title":"PVLDB"},{"key":"782_CR44","doi-asserted-by":"publisher","unstructured":"Wang, K., Lin, X., Qin, L., Zhang, W., Zhang, Y.: Efficient bitruss decomposition for large-scale bipartite graphs. In: ICDE, pp. 661\u2013672 (2020). https:\/\/doi.org\/10.1109\/ICDE48307.2020.00063","DOI":"10.1109\/ICDE48307.2020.00063"},{"issue":"9","key":"782_CR45","first-page":"1889","volume":"15","author":"Y Wang","year":"2022","unstructured":"Wang, Y., Xu, R., Jian, X., Zhou, A., Chen, L.: Towards distributed bitruss decomposition on bipartite graphs. PVLDB 15(9), 1889\u20131901 (2022)","journal-title":"PVLDB"},{"key":"782_CR46","first-page":"1305","volume":"16","author":"B Wilder","year":"2017","unstructured":"Wilder, B., Yadav, A., Immorlica, N., Rice, E., Tambe, M.: Uncharted but not uninfluenced: influence maximization with an uncertain network. AAMAS 16, 1305\u20131313 (2017)","journal-title":"AAMAS"},{"key":"782_CR47","doi-asserted-by":"publisher","unstructured":"Wolf, M.M., Deveci, M., Berry, J.W., Hammond, S.D., Rajamanickam, S.: Fast linear algebra-based triangle counting with kokkoskernels. In: HPEC, pp. 1\u20137 (2017). https:\/\/doi.org\/10.1109\/HPEC.2017.8091043","DOI":"10.1109\/HPEC.2017.8091043"},{"key":"782_CR48","doi-asserted-by":"crossref","unstructured":"Yuan, Y., Chen, L., Wang, G.: Efficiently answering probability threshold-based shortest path queries over uncertain graphs. In: DASFAA, pp. 155\u2013170 (2010)","DOI":"10.1007\/978-3-642-12026-8_14"},{"issue":"12","key":"782_CR49","doi-asserted-by":"publisher","first-page":"2767","DOI":"10.1109\/TKDE.2012.222","volume":"25","author":"Y Yuan","year":"2013","unstructured":"Yuan, Y., Wang, G., Chen, L., Wang, H.: Efficient keyword search on uncertain graph data. TKDE 25(12), 2767\u20132779 (2013). https:\/\/doi.org\/10.1109\/TKDE.2012.222","journal-title":"TKDE"},{"issue":"3","key":"782_CR50","first-page":"320","volume":"13","author":"Y Zeng","year":"2020","unstructured":"Zeng, Y., Tong, Y., Chen, L.: Last-mile delivery made practical: an efficient route planning framework with theoretical guarantees. PVLDB 13(3), 320\u2013333 (2020)","journal-title":"PVLDB"},{"issue":"1","key":"782_CR51","doi-asserted-by":"publisher","first-page":"110","DOI":"10.1186\/1471-2105-15-110","volume":"15","author":"Y Zhang","year":"2014","unstructured":"Zhang, Y., Phillips, C.A., Rogers, G.L., Baker, E.J., Chesler, E.J., Langston, M.A.: On finding bicliques in bipartite graphs: a novel algorithm and its application to the integration of diverse biological data types. BMC Bioinform. 15(1), 110 (2014). https:\/\/doi.org\/10.1186\/1471-2105-15-110","journal-title":"BMC Bioinform."},{"issue":"3","key":"782_CR52","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1109\/TCBB.2013.2297915","volume":"11","author":"B Zhao","year":"2014","unstructured":"Zhao, B., Wang, J., Li, M., Wu, F., Pan, Y.: Detecting protein complexes based on uncertain graph model. IEEE\/ACM Trans. Comput. Biol. Bioinform. 11(3), 486\u2013497 (2014). https:\/\/doi.org\/10.1109\/TCBB.2013.2297915","journal-title":"IEEE\/ACM Trans. Comput. Biol. Bioinform."},{"issue":"12","key":"782_CR53","doi-asserted-by":"publisher","first-page":"2576","DOI":"10.14778\/3407790.3407846","volume":"13","author":"A Zhou","year":"2020","unstructured":"Zhou, A., Wang, Y., Chen, L.: Finding large diverse communities on networks: the edge maximum k*-partite clique. PVLDB 13(12), 2576\u20132589 (2020). https:\/\/doi.org\/10.14778\/3407790.3407846","journal-title":"PVLDB"},{"issue":"2","key":"782_CR54","doi-asserted-by":"publisher","first-page":"211","DOI":"10.14778\/3489496.3489502","volume":"15","author":"A Zhou","year":"2021","unstructured":"Zhou, A., Wang, Y., Chen, L.: Butterfly counting on uncertain bipartite graphs. PVLDB 15(2), 211\u2013223 (2021). https:\/\/doi.org\/10.14778\/3489496.3489502","journal-title":"PVLDB"},{"key":"782_CR55","doi-asserted-by":"publisher","first-page":"046115","DOI":"10.1103\/PhysRevE.76.046115","volume":"76","author":"T Zhou","year":"2007","unstructured":"Zhou, T., Ren, J., Medo, M.C.V., Zhang, Y.C.: Bipartite network projection and personal recommendation. Phys. Rev. E 76, 046115 (2007). https:\/\/doi.org\/10.1103\/PhysRevE.76.046115","journal-title":"Phys. Rev. E"},{"key":"782_CR56","doi-asserted-by":"crossref","unstructured":"Zou, Z.: Bitruss decomposition of bipartite graphs. In: DASFAA, pp. 218\u2013233 (2016)","DOI":"10.1007\/978-3-319-32049-6_14"},{"issue":"1","key":"782_CR57","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/s10115-016-0943-y","volume":"50","author":"Z Zou","year":"2017","unstructured":"Zou, Z., Zhu, R.: Truss decomposition of uncertain graphs. Knowl. Inf. Syst. 50(1), 197\u2013230 (2017)","journal-title":"Knowl. Inf. Syst."}],"container-title":["The VLDB Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00778-023-00782-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00778-023-00782-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00778-023-00782-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,15]],"date-time":"2023-08-15T10:11:30Z","timestamp":1692094290000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00778-023-00782-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,17]]},"references-count":57,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["782"],"URL":"https:\/\/doi.org\/10.1007\/s00778-023-00782-4","relation":{},"ISSN":["1066-8888","0949-877X"],"issn-type":[{"value":"1066-8888","type":"print"},{"value":"0949-877X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,17]]},"assertion":[{"value":"29 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}