{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T05:45:49Z","timestamp":1782971149436,"version":"3.54.5"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T00:00:00Z","timestamp":1621987200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T00:00:00Z","timestamp":1621987200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Paris Lodron University of Salzburg"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Distrib. Comput."],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The <jats:italic>2-Choices<\/jats:italic> dynamics is a process that models voting behavior on networks and works as follows: Each agent initially holds either opinion <jats:italic>blue<\/jats:italic> or <jats:italic>red<\/jats:italic>; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. We study its behavior on a class of networks with core\u2013periphery structure. Assume that a densely-connected subset of agents, the <jats:italic>core<\/jats:italic>, holds a different opinion from the rest of the network, the <jats:italic>periphery<\/jats:italic>. We prove that, depending on the strength of the cut between core and periphery, a phase-transition phenomenon occurs: Either the core\u2019s opinion rapidly spreads across the network, or a <jats:italic>metastability<\/jats:italic> phase takes place in which both opinions coexist for superpolynomial time. The interest of our result, which we also validate with extensive experiments on real networks, is twofold. First, it sheds light on the <jats:italic>influence<\/jats:italic> of the core on the rest of the network as a function of its connectivity toward the latter. Second, it is one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network.\n<\/jats:p>","DOI":"10.1007\/s00446-021-00396-5","type":"journal-article","created":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T05:02:42Z","timestamp":1622005362000},"page":"207-225","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Phase transition of the 2-Choices dynamics on core\u2013periphery networks"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4744-5635","authenticated-orcid":false,"given":"Emilio","family":"Cruciani","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emanuele","family":"Natale","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andr\u00e9","family":"Nusser","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Giacomo","family":"Scornavacca","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,5,26]]},"reference":[{"key":"396_CR1","unstructured":"Aldous, D., Fill, J.A.: Reversible Markov chains and random walks on graphs. Unfinished monograph, recompiled 2014. https:\/\/www.stat.berkeley.edu\/users\/aldous\/RWG\/book.html (2002)"},{"key":"396_CR2","doi-asserted-by":"publisher","unstructured":"Anagnostopoulos, A., Becchetti, L., Cruciani, E., Pasquale, F., Rizzo, S.: Biased opinion dynamics: when the devil is in the details. In: Bessiere, C. (ed.) Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI 2020, pp. 53\u201359. ijcai.org (2020). https:\/\/doi.org\/10.24963\/ijcai.2020\/8","DOI":"10.24963\/ijcai.2020\/8"},{"key":"396_CR3","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1016\/j.jpdc.2016.08.003","volume":"99","author":"C Avin","year":"2017","unstructured":"Avin, C., Borokhovich, M., Lotker, Z., Peleg, D.: Distributed computing on core\u2013periphery networks: axiom-based design. J. Parallel Distrib. Comput. 99, 51\u201367 (2017). https:\/\/doi.org\/10.1016\/j.jpdc.2016.08.003","journal-title":"J. Parallel Distrib. Comput."},{"key":"396_CR4","doi-asserted-by":"publisher","unstructured":"Avin, C., Lotker, Z., Mizrachi, A., Peleg, D.: Majority vote and monopolies in social networks. In: Proceedings of the 20th International Conference on Distributed Computing and Networking, ICDCN \u201919, p. 342\u2013351. Association for Computing Machinery, New York, NY, USA (2019). https:\/\/doi.org\/10.1145\/3288599.3288633","DOI":"10.1145\/3288599.3288633"},{"issue":"10","key":"396_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pone.0205820","volume":"13","author":"C Avin","year":"2018","unstructured":"Avin, C., Lotker, Z., Peleg, D., Pignolet, Y.A., Turkel, I.: Elites in social networks: an axiomatic approach to power balance and price\u2019s square root law. PLoS One 13(10), 1\u201335 (2018). https:\/\/doi.org\/10.1371\/journal.pone.0205820","journal-title":"PLoS One"},{"issue":"4","key":"396_CR6","doi-asserted-by":"publisher","first-page":"624","DOI":"10.1007\/s00440-005-0451-6","volume":"134","author":"J Balogh","year":"2006","unstructured":"Balogh, J., Bollob\u00e1s, B.: Bootstrap percolation on the hypercube. Probab. Theory Relat. Fields 134(4), 624\u2013648 (2006)","journal-title":"Probab. Theory Relat. Fields"},{"issue":"1\u20132","key":"396_CR7","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1002\/rsa.20158","volume":"30","author":"J Balogh","year":"2007","unstructured":"Balogh, J., Pittel, B.G.: Bootstrap percolation on the random regular graph. Random Struct. Algorithms 30(1\u20132), 257\u2013286 (2007). https:\/\/doi.org\/10.1002\/rsa.20158","journal-title":"Random Struct. Algorithms"},{"issue":"1","key":"396_CR8","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1145\/3388392.3388403","volume":"51","author":"L Becchetti","year":"2020","unstructured":"Becchetti, L., Clementi, A., Natale, E.: Consensus dynamics: an overview. SIGACT News 51(1), 58\u2013104 (2020). https:\/\/doi.org\/10.1145\/3388392.3388403","journal-title":"SIGACT News"},{"key":"396_CR9","unstructured":"Becchetti, L., Clementi, A., Natale, E., Pasquale, F., Silvestri, R.: Plurality consensus in the gossip model. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA\u201915), pp. 371\u2013390. SIAM (2015). http:\/\/dl.acm.org\/citation.cfm?id=2722129.2722156"},{"issue":"4","key":"396_CR10","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s00446-016-0289-4","volume":"30","author":"L Becchetti","year":"2017","unstructured":"Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Silvestri, R., Trevisan, L.: Simple dynamics for plurality consensus. Distrib. Comput. 30(4), 293\u2013306 (2017). https:\/\/doi.org\/10.1007\/s00446-016-0289-4","journal-title":"Distrib. Comput."},{"key":"396_CR11","doi-asserted-by":"publisher","unstructured":"Becchetti, L., Clementi, A.E.F., Natale, E., Pasquale, F., Trevisan, L.: Stabilizing consensus with many opinions. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10\u201312, 2016, pp. 620\u2013635 (2016). https:\/\/doi.org\/10.1137\/1.9781611974331.ch46","DOI":"10.1137\/1.9781611974331.ch46"},{"issue":"9","key":"396_CR12","doi-asserted-by":"publisher","first-page":"2719","DOI":"10.1016\/j.spa.2016.02.015","volume":"126","author":"I Benjamini","year":"2016","unstructured":"Benjamini, I., Chan, S.O., O\u2019Donnell, R., Tamuz, O., Tan, L.Y.: Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs. Stoch. Process. Appl. 126(9), 2719\u20132733 (2016). https:\/\/doi.org\/10.1016\/j.spa.2016.02.015","journal-title":"Stoch. Process. Appl."},{"key":"396_CR13","doi-asserted-by":"publisher","unstructured":"Berenbrink, P., Friedetzky, T., Giakkoupis, G., Kling, P.: Efficient plurality consensus, or: the benefits of cleaning up from time to time. In: 43rd International Colloquium on Automata, Languages and Programming (ICALP 2016). Rome, Italy (2016). https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.136, https:\/\/hal.archives-ouvertes.fr\/hal-01353690","DOI":"10.4230\/LIPIcs.ICALP.2016.136"},{"key":"396_CR14","doi-asserted-by":"publisher","unstructured":"Berenbrink, P., Giakkoupis, G., Kermarrec, A.M., Mallmann-Trenn, F.: Bounds on the voter model in dynamic networks. In: Chatzigiannakis, I., Mitzenmacher, M., Rabani, Y., Sangiorgi, D. (eds.) 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), Leibniz International Proceedings in Informatics (LIPIcs), vol.\u00a055, pp. 146:1\u2013146:15. Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2016). https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.146, http:\/\/drops.dagstuhl.de\/opus\/volltexte\/2016\/6290","DOI":"10.4230\/LIPIcs.ICALP.2016.146"},{"issue":"2","key":"396_CR15","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1006\/jctb.2001.2045","volume":"83","author":"E Berger","year":"2001","unstructured":"Berger, E.: Dynamic monopolies of constant size. J. Comb. Theory Ser. B 83(2), 191\u2013200 (2001). https:\/\/doi.org\/10.1006\/jctb.2001.2045","journal-title":"J. Comb. Theory Ser. B"},{"issue":"4","key":"396_CR16","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1016\/S0378-8733(99)00019-2","volume":"21","author":"SP Borgatti","year":"2000","unstructured":"Borgatti, S.P., Everett, M.G.: Models of core\/periphery structures. Soc. Netw. 21(4), 375\u2013395 (2000). https:\/\/doi.org\/10.1016\/S0378-8733(99)00019-2","journal-title":"Soc. Netw."},{"key":"396_CR17","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1007\/3-540-44436-X_10","volume-title":"Greedy Approximation Algorithms for Finding Dense Components in a Graph","author":"M Charikar","year":"2000","unstructured":"Charikar, M.: Greedy Approximation Algorithms for Finding Dense Components in a Graph, pp. 84\u201395. Springer, Berlin (2000). https:\/\/doi.org\/10.1007\/3-540-44436-X_10"},{"key":"396_CR18","doi-asserted-by":"publisher","unstructured":"Cooper, C., Dyer, M.E., Frieze, A.M., Rivera, N.: Discordant voting processes on finite graphs. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11\u201315, 2016, Rome, Italy, pp. 145:1\u2013145:13 (2016). https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.145","DOI":"10.4230\/LIPIcs.ICALP.2016.145"},{"key":"396_CR19","doi-asserted-by":"publisher","unstructured":"Cooper, C., Els\u00e4sser, R., Radzik, T.: The power of two choices in distributed voting. In: Automata, Languages, and Programming\u201441st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8\u201311, 2014, Proceedings, Part II, pp. 435\u2013446 (2014). https:\/\/doi.org\/10.1007\/978-3-662-43951-7_37","DOI":"10.1007\/978-3-662-43951-7_37"},{"key":"396_CR20","doi-asserted-by":"publisher","unstructured":"Cooper, C., Els\u00e4sser, R., Radzik, T., Rivera, N., Shiraga, T.: Fast consensus for voting on general expander graphs. In: Distributed Computing\u201429th International Symposium, DISC 2015, Tokyo, Japan, October 7\u20139, 2015, Proceedings, pp. 248\u2013262 (2015). https:\/\/doi.org\/10.1007\/978-3-662-48653-5_17","DOI":"10.1007\/978-3-662-48653-5_17"},{"key":"396_CR21","doi-asserted-by":"publisher","unstructured":"Cooper, C., Radzik, T., Rivera, N.: The coalescing-branching random walk on expanders and the dual epidemic process. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25\u201328, 2016, pp. 461\u2013467 (2016). https:\/\/doi.org\/10.1145\/2933057.2933119","DOI":"10.1145\/2933057.2933119"},{"key":"396_CR22","doi-asserted-by":"publisher","unstructured":"Cooper, C., Radzik, T., Rivera, N., Shiraga, T.: Fast plurality consensus in regular expanders. In: 31st International Symposium on Distributed Computing, DISC, October 16\u201320, 2017, Vienna, Austria, pp. 13:1\u201313:16 (2017). https:\/\/doi.org\/10.4230\/LIPIcs.DISC.2017.13","DOI":"10.4230\/LIPIcs.DISC.2017.13"},{"key":"396_CR23","doi-asserted-by":"publisher","unstructured":"Cooper, C., Rivera, N.: The linear voting model. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11\u201315, 2016, Rome, Italy, pp. 144:1\u2013144:12 (2016). https:\/\/doi.org\/10.4230\/LIPIcs.ICALP.2016.144","DOI":"10.4230\/LIPIcs.ICALP.2016.144"},{"key":"396_CR24","doi-asserted-by":"publisher","unstructured":"Cruciani, E., Mimun, H.A., Quattropani, M., Rizzo, S.: Brief announcement: phase transitions of the k-majority dynamics in a biased communication model. In: 34th International Symposium on Distributed Computing, DISC 2020, LIPIcs, vol. 179, pp. 42:1\u201342:3. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik (2020). https:\/\/doi.org\/10.4230\/LIPIcs.DISC.2020.42","DOI":"10.4230\/LIPIcs.DISC.2020.42"},{"key":"396_CR25","doi-asserted-by":"publisher","unstructured":"Cruciani, E., Mimun, H.A., Quattropani, M., Rizzo, S.: Phase transitions of the k-majority dynamics in a biased communication model. In: ICDCN \u201921: International Conference on Distributed Computing and Networking, 2021, pp. 146\u2013155. ACM (2021). https:\/\/doi.org\/10.1145\/3427796.3427811","DOI":"10.1145\/3427796.3427811"},{"key":"396_CR26","unstructured":"Cruciani, E., Natale, E., Nusser, A., Scornavacca, G.: Phase transition of the 2-choices dynamics on core\u2013periphery networks. In: Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS 2018, pp. 777\u2013785 (2018). http:\/\/dl.acm.org\/citation.cfm?id=3237499"},{"key":"396_CR27","doi-asserted-by":"publisher","unstructured":"Cruciani, E., Natale, E., Scornavacca, G.: Distributed community detection via metastability of the 2-choices dynamics. In: The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, pp. 6046\u20136053. AAAI Press (2019). https:\/\/doi.org\/10.1609\/aaai.v33i01.33016046","DOI":"10.1609\/aaai.v33i01.33016046"},{"key":"396_CR28","volume-title":"Networks, Crowds, and Markets: Reasoning About a Highly Connected World","author":"E David","year":"2010","unstructured":"David, E., Jon, K.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York, USA (2010)"},{"issue":"345","key":"396_CR29","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1080\/01621459.1974.10480137","volume":"69","author":"MH Degroot","year":"1974","unstructured":"Degroot, M.H.: Reaching a consensus. J. Am. Stat. Assoc. 69(345), 118\u2013121 (1974). https:\/\/doi.org\/10.1080\/01621459.1974.10480137","journal-title":"J. Am. Stat. Assoc."},{"key":"396_CR30","doi-asserted-by":"publisher","unstructured":"Doerr, B., Goldberg, L.A., Minder, L., Sauerwald, T., Scheideler, C.: Stabilizing consensus with the power of two choices. In: Proceedings of the Twenty-third Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA \u201911, pp. 149\u2013158. ACM, New York, NY, USA (2011). https:\/\/doi.org\/10.1145\/1989493.1989516","DOI":"10.1145\/1989493.1989516"},{"key":"396_CR31","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581274","volume-title":"Concentration of Measure for the Analysis of Randomized Algorithms","author":"D Dubhashi","year":"2009","unstructured":"Dubhashi, D., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms, 1st edn. Cambridge University Press, New York (2009)","edition":"1"},{"key":"396_CR32","doi-asserted-by":"crossref","unstructured":"French, J.R.: A formal theory of social power. Psychol. Rev. 63(3), 181\u2013194 (1956-05)","DOI":"10.1037\/h0046123"},{"key":"396_CR33","doi-asserted-by":"crossref","unstructured":"Ghaffari, M., Lengler, J.: Tight analysis for the 3-majority consensus dynamics. CoRR arXiv:1705.05583 (2017)","DOI":"10.1145\/3212734.3212738"},{"key":"396_CR34","doi-asserted-by":"publisher","unstructured":"Ghaffari, M., Parter, M.: A polylogarithmic gossip algorithm for plurality consensus. In: Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing, PODC 2016, Chicago, IL, USA, July 25\u201328, 2016, pp. 117\u2013126 (2016). https:\/\/doi.org\/10.1145\/2933057.2933097","DOI":"10.1145\/2933057.2933097"},{"issue":"1","key":"396_CR35","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1257\/mic.2.1.112","volume":"2","author":"B Golub","year":"2010","unstructured":"Golub, B., Jackson, M.O.: Na\u00efve learning in social networks and the wisdom of crowds. Am. Econ. J. Microecon. 2(1), 112\u201349 (2010). https:\/\/doi.org\/10.1257\/mic.2.1.112","journal-title":"Am. Econ. J. Microecon."},{"key":"396_CR36","unstructured":"Harary, F.: A criterion for unanimity in French\u2019s theory of social power. Studies in Social Power, pp. 168\u2013182 (1959)"},{"issue":"2","key":"396_CR37","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1006\/inco.2001.3088","volume":"171","author":"Y Hassin","year":"2001","unstructured":"Hassin, Y., Peleg, D.: Distributed probabilistic polling and applications to proportionate agreement. Inf. Comput. 171(2), 248\u2013268 (2001). https:\/\/doi.org\/10.1006\/inco.2001.3088","journal-title":"Inf. Comput."},{"issue":"4","key":"396_CR38","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/S0036144500371907","volume":"42","author":"HW Hethcote","year":"2000","unstructured":"Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42(4), 599\u2013653 (2000). https:\/\/doi.org\/10.1137\/S0036144500371907","journal-title":"SIAM Rev."},{"issue":"4","key":"396_CR39","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1090\/S0273-0979-06-01126-8","volume":"43","author":"S Hoory","year":"2006","unstructured":"Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439\u2013561 (2006)","journal-title":"Bull. Am. Math. Soc."},{"key":"396_CR40","unstructured":"Jackson, M.O.: Social and economic networks. Princeton University Press, Princeton (2010-11-21)"},{"issue":"5","key":"396_CR41","doi-asserted-by":"publisher","first-page":"1989","DOI":"10.1214\/11-AAP822","volume":"22","author":"S Janson","year":"2012","unstructured":"Janson, S., \u0141uczak, T., Turova, T., Vallier, T.: Bootstrap percolation on the random graph $$G_{n, p}$$. Ann. Appl. Probab. 22(5), 1989\u20132047 (2012). https:\/\/doi.org\/10.1214\/11-AAP822","journal-title":"Ann. Appl. Probab."},{"issue":"5","key":"396_CR42","doi-asserted-by":"publisher","first-page":"1694","DOI":"10.1214\/10-AAP729","volume":"21","author":"Y Kanoria","year":"2011","unstructured":"Kanoria, Y., Montanari, A.: Majority dynamics on trees and the dynamic cavity method. Ann. Appl. Probab. 21(5), 1694\u20131748 (2011). https:\/\/doi.org\/10.1214\/10-AAP729","journal-title":"Ann. Appl. Probab."},{"issue":"1","key":"396_CR43","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1038\/s41598-020-79139-8","volume":"11","author":"T Kobayashi","year":"2021","unstructured":"Kobayashi, T., G\u00e9nois, M.: The switching mechanisms of social network densification. Sci. Rep. 11(1), 1\u201311 (2021)","journal-title":"Sci. Rep."},{"key":"396_CR44","doi-asserted-by":"crossref","unstructured":"Kunegis, J.: Konect: the koblenz network collection. In: Proceedings of the 22nd International Conference on World Wide Web, pp. 1343\u20131350. ACM (2013)","DOI":"10.1145\/2487788.2488173"},{"key":"396_CR45","doi-asserted-by":"publisher","first-page":"030101","DOI":"10.1103\/PhysRevE.75.030101","volume":"75","author":"R Lambiotte","year":"2007","unstructured":"Lambiotte, R., Ausloos, M., Ho\u0142yst, J.A.: Majority model on a network with communities. Phys. Rev. E 75, 030101 (2007). https:\/\/doi.org\/10.1103\/PhysRevE.75.030101","journal-title":"Phys. Rev. E"},{"key":"396_CR46","unstructured":"Leskovec, J., Krevl, A.: SNAP Datasets: Stanford large network dataset collection. http:\/\/snap.stanford.edu\/data (2014)"},{"key":"396_CR47","doi-asserted-by":"publisher","DOI":"10.1090\/mbk\/107","volume-title":"Markov Chains and Mixing Times","author":"DA Levin","year":"2017","unstructured":"Levin, D.A., Peres, Y.: Markov Chains and Mixing Times, vol. 107. American Mathematical Society, Providence (2017)"},{"key":"396_CR48","unstructured":"Liggett, T.M.: Interacting Particle Systems. Springer Science & Business Media (2012-12-06). Google-Books-ID: 7JbqBwAAQBAJ"},{"key":"396_CR49","unstructured":"Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer Science & Business Media (2013-03-09). Google-Books-ID: wRv2CAAAQBAJ"},{"issue":"3","key":"396_CR50","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1371\/journal.pone.0119678","volume":"10","author":"A Ma","year":"2015","unstructured":"Ma, A., Mondrag\u00f3n, R.J.: Rich-cores in networks. PLoS One 10(3), 1\u201313 (2015). https:\/\/doi.org\/10.1371\/journal.pone.0119678","journal-title":"PLoS One"},{"issue":"3","key":"396_CR51","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1007\/s10458-013-9230-4","volume":"28","author":"E Mossel","year":"2014","unstructured":"Mossel, E., Neeman, J., Tamuz, O.: Majority dynamics and aggregation of information in social networks. Auton. Agents Multi-Agent Syst. 28(3), 408\u2013429 (2014). https:\/\/doi.org\/10.1007\/s10458-013-9230-4","journal-title":"Auton. Agents Multi-Agent Syst."},{"key":"396_CR52","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1214\/14-PS230","volume":"14","author":"E Mossel","year":"2017","unstructured":"Mossel, E., Tamuz, O.: Opinion exchange dynamics. Probab. Surv. 14, 155\u2013204 (2017). https:\/\/doi.org\/10.1214\/14-PS230","journal-title":"Probab. Surv."},{"key":"396_CR53","doi-asserted-by":"publisher","first-page":"016128","DOI":"10.1103\/PhysRevE.66.016128","volume":"66","author":"MEJ Newman","year":"2002","unstructured":"Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E Stat. Nonlinear Soft Matter phys. 66, 016128 (2002)","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter phys."},{"issue":"2\u20133","key":"396_CR54","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1016\/S0166-218X(98)00043-2","volume":"86","author":"D Peleg","year":"1998","unstructured":"Peleg, D.: Size bounds for dynamic monopolies. Discrete Appl. Math. 86(2\u20133), 263\u2013273 (1998). https:\/\/doi.org\/10.1016\/S0166-218X(98)00043-2","journal-title":"Discrete Appl. Math."},{"issue":"2","key":"396_CR55","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/S0304-3975(01)00055-X","volume":"282","author":"D Peleg","year":"2002","unstructured":"Peleg, D.: Local majorities, coalitions and monopolies in graphs: a review. Theor. Comput. Sci. 282(2), 231\u2013257 (2002). https:\/\/doi.org\/10.1016\/S0304-3975(01)00055-X","journal-title":"Theor. Comput. Sci."},{"issue":"1","key":"396_CR56","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1137\/120881683","volume":"74","author":"MP Rombach","year":"2014","unstructured":"Rombach, M.P., Porter, M.A., Fowler, J.H., Mucha, P.J.: Core-periphery structure in networks. SIAM J. Appl. Math. 74(1), 167\u2013190 (2014). https:\/\/doi.org\/10.1137\/120881683","journal-title":"SIAM J. Appl. Math."},{"key":"396_CR57","doi-asserted-by":"publisher","unstructured":"Shimizu, N., Shiraga, T.: Phase transitions of best-of-two and best-of-three on stochastic block models. In: J.\u00a0Suomela (ed.) 33rd International Symposium on Distributed Computing, DISC 2019, October 14\u201318, 2019, Budapest, Hungary, LIPIcs, vol. 146, pp. 32:1\u201332:17. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik (2019). https:\/\/doi.org\/10.4230\/LIPIcs.DISC.2019.32","DOI":"10.4230\/LIPIcs.DISC.2019.32"},{"issue":"5","key":"396_CR58","doi-asserted-by":"publisher","first-page":"1096","DOI":"10.1086\/226902","volume":"84","author":"D Snyder","year":"1979","unstructured":"Snyder, D., Kick, E.L.: Structural position in the world system and economic growth, 1955\u20131970: a multiple-network analysis of transnational interactions. Am. J. Sociol. 84(5), 1096\u20131126 (1979)","journal-title":"Am. J. Sociol."},{"key":"396_CR59","doi-asserted-by":"publisher","unstructured":"Zehmakan, A.N.: Two phase transitions in two-way bootstrap percolation. In: Lu, P., Zhang, G. (eds.) 30th International Symposium on Algorithms and Computation (ISAAC 2019), Leibniz International Proceedings in Informatics (LIPIcs), vol. 149, pp. 5:1\u20135:21. Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2019). https:\/\/doi.org\/10.4230\/LIPIcs.ISAAC.2019.5. https:\/\/drops.dagstuhl.de\/opus\/volltexte\/2019\/11501","DOI":"10.4230\/LIPIcs.ISAAC.2019.5"},{"key":"396_CR60","doi-asserted-by":"publisher","first-page":"032803","DOI":"10.1103\/PhysRevE.91.032803","volume":"91","author":"X Zhang","year":"2015","unstructured":"Zhang, X., Martin, T., Newman, M.E.J.: Identification of core-periphery structure in networks. Phys. Rev. E 91, 032803 (2015). https:\/\/doi.org\/10.1103\/PhysRevE.91.032803","journal-title":"Phys. Rev. E"},{"issue":"3","key":"396_CR61","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1109\/LCOMM.2004.823426","volume":"8","author":"S Zhou","year":"2004","unstructured":"Zhou, S., Mondrag\u00f3n, R.J.: The rich-club phenomenon in the internet topology. IEEE Commun. Lett. 8(3), 180\u2013182 (2004)","journal-title":"IEEE Commun. Lett."}],"container-title":["Distributed Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00446-021-00396-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00446-021-00396-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00446-021-00396-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,19]],"date-time":"2021-06-19T04:21:05Z","timestamp":1624076465000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00446-021-00396-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,26]]},"references-count":61,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["396"],"URL":"https:\/\/doi.org\/10.1007\/s00446-021-00396-5","relation":{},"ISSN":["0178-2770","1432-0452"],"issn-type":[{"value":"0178-2770","type":"print"},{"value":"1432-0452","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,26]]},"assertion":[{"value":"3 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 May 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}