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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,17]],"date-time":"2024-02-17T00:08:29Z","timestamp":1708128509726},"reference-count":32,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T00:00:00Z","timestamp":1685059200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2024,3]]},"abstract":"Abstract<\/jats:title>We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with a type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge e<\/jats:italic> is given by a random variable

\n$X_1(e)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (resp.

\n$X_2(e)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent

\n$\\tau \\in (1,2)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, we show that with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage-time distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than

\n$n^\\alpha$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for some

\n$\\alpha\\gt 0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/jpr.2023.23","type":"journal-article","created":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T06:27:48Z","timestamp":1685082468000},"page":"137-152","update-policy":"http:\/\/dx.doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["The winner takes it all but one"],"prefix":"10.1017","volume":"61","author":[{"given":"Maria","family":"Deijfen","sequence":"first","affiliation":[]},{"given":"Remco","family":"van der Hofstad","sequence":"additional","affiliation":[]},{"given":"Matteo","family":"Sfragara","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2023,5,26]]},"reference":[{"key":"S0021900223000232_ref14","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-019-08746-5"},{"key":"S0021900223000232_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(00)00042-9"},{"key":"S0021900223000232_ref6","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v20-3749"},{"key":"S0021900223000232_ref27","unstructured":"[27] Van der Hofstad, R. (2023). Random Graphs and Complex Networks, volume 2. Available at https:\/\/www.win.tue.nl\/~rhofstad\/NotesRGCNII.pdf."},{"key":"S0021900223000232_ref11","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1282924060"},{"key":"S0021900223000232_ref16","doi-asserted-by":"publisher","DOI":"10.1214\/15-AAP1151"},{"key":"S0021900223000232_ref28","unstructured":"[28] Van der Hofstad, R. and Komj\u00e1thy, J. (2015). Fixed speed competition on the configuration model with infinite variance degrees: equal speeds. Available at arXiv:1503.09046."},{"key":"S0021900223000232_ref29","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20063"},{"key":"S0021900223000232_ref30","unstructured":"[30] Van der Hofstad, R. , Hooghiemstra, G. and Znamenski, D. (2005). Random graphs with arbitrary i.i.d. degrees. Available at arXiv:math\/0502580."},{"key":"S0021900223000232_ref23","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240060204"},{"key":"S0021900223000232_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/10-AAP753"},{"key":"S0021900223000232_ref20","doi-asserted-by":"crossref","unstructured":"[20] Hammersley, J.M. and Welsh, D.J.A. (1965). First passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli 1713, Bayes 1763, Laplace 1813, ed. J. Neyman and L. M. Le Cam, pp. 61\u2013110. Springer.","DOI":"10.1007\/978-3-642-49750-6_7"},{"key":"S0021900223000232_ref1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20846"},{"key":"S0021900223000232_ref22","unstructured":"[22] Janson, S. (2002). On concentration of probability. In Contemporary Combinatorics (Bolyai Society Mathematical Studies 10), pp. 289\u2013301. Springer, Berlin and Heidelberg."},{"key":"S0021900223000232_ref12","doi-asserted-by":"publisher","DOI":"10.1214\/16-AOP1120"},{"key":"S0021900223000232_ref7","doi-asserted-by":"publisher","DOI":"10.1017\/jpr.2016.92"},{"key":"S0021900223000232_ref10","doi-asserted-by":"publisher","DOI":"10.1214\/09-AAP666"},{"key":"S0021900223000232_ref17","doi-asserted-by":"publisher","DOI":"10.1214\/105051604000000503"},{"key":"S0021900223000232_ref31","first-page":"703","article-title":"Distances in random graphs with finite mean and infinite variance degrees","volume":"12","author":"Van der Hofstad","year":"2005","journal-title":"Electron. J. 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