{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:00:23Z","timestamp":1753884023039,"version":"3.41.2"},"reference-count":53,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Mod. Phys. C"],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:p> We present an alternative finite-size scaling (FSS) of the contact process on scale-free networks compatible with mean-field scaling and test it with extensive Monte Carlo simulations. In our FSS theory, the dependence on the system size enters the external field, which represents spontaneous contamination in the context of an epidemic model. In addition, dependence on the finite size in the scale-free networks also enters the network cutoff. We show that our theory reproduces the results of other mean-field theories on finite lattices already reported in the literature. To simulate the dynamics, we impose quasi-stationary states by reactivation. We insert spontaneously infected individuals, equivalent to a droplet perturbation to the system scaling as [Formula: see text]. The system presents an absorbing phase transition where the critical behavior obeys the mean-field exponents, as we show theoretically and by simulations. However, the quasi-stationary state gives finite-size logarithmic corrections, predicted by our FSS theory, and reproduces equivalent results in the literature in the thermodynamic limit. We also report the critical threshold estimates of basic reproduction number [Formula: see text] of the model as a linear function of the network connectivity inverse [Formula: see text], and the extrapolation of the critical threshold function for [Formula: see text] yields the basic reproduction number [Formula: see text] of the complete graph, as expected. Decreasing the network connectivity increases the critical [Formula: see text] for this model. <\/jats:p>","DOI":"10.1142\/s012918312350105x","type":"journal-article","created":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T11:49:15Z","timestamp":1671536955000},"source":"Crossref","is-referenced-by-count":1,"title":["Droplet finite-size scaling of the contact process on scale-free networks revisited"],"prefix":"10.1142","volume":"34","author":[{"given":"D. S. M.","family":"Alencar","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Piau\u00ed, 57072-970 Teresina, Piau\u00ed, Brazil"}]},{"given":"T. F. A.","family":"Alves","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Piau\u00ed, 57072-970 Teresina, Piau\u00ed, Brazil"}]},{"given":"R. S.","family":"Ferreira","sequence":"additional","affiliation":[{"name":"Departamento de Ci\u00eancias Exatas e Aplicadas,  Universidade Federal de Ouro Preto, 35931-008 Jo\u00e3o Monlevade, MG, Brazil"}]},{"given":"G. A.","family":"Alves","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual do Piau\u00ed, 64002-150 Teresina, PI, Brazil"}]},{"given":"A.","family":"Macedo-Filho","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Estadual do Piau\u00ed, 64002-150 Teresina, PI, Brazil"}]},{"given":"F. W. S.","family":"Lima","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica, Universidade Federal do Piau\u00ed, 57072-970 Teresina, PI, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2023,2,13]]},"reference":[{"key":"S012918312350105XBIB001","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780198515906.001.0001","volume-title":"Evolution of Networks: From Biological Nets to the Internet and WWW","author":"Dorogovtsev S. N.","year":"2003"},{"volume-title":"Network Science","year":"2016","author":"Barab\u00e1si A.-L.","key":"S012918312350105XBIB002"},{"key":"S012918312350105XBIB003","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1038\/35082140","volume":"411","author":"Liljeros F.","year":"2001","journal-title":"Nature"},{"key":"S012918312350105XBIB004","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","volume":"286","author":"Barab\u00e1si A.-L.","year":"1999","journal-title":"Science"},{"key":"S012918312350105XBIB005","doi-asserted-by":"crossref","first-page":"3747","DOI":"10.1073\/pnas.0400087101","volume":"101","author":"Barrat A.","year":"2004","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"S012918312350105XBIB006","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1126\/science.149.3683.510","volume":"149","author":"de Solla Price D. J.","year":"1965","journal-title":"Science"},{"key":"S012918312350105XBIB007","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s100510050359","volume":"4","author":"Redner S.","year":"1998","journal-title":"Eur. Phys. J. B"},{"key":"S012918312350105XBIB008","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1073\/pnas.98.2.404","volume":"98","author":"Newman M. E. J.","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"S012918312350105XBIB009","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/S0378-4371(02)00736-7","volume":"311","author":"Barab\u00e1si A.-L.","year":"2002","journal-title":"Physica A"},{"key":"S012918312350105XBIB010","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","volume":"74","author":"Albert R.","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"S012918312350105XBIB011","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","volume":"45","author":"Newman M. E. J.","year":"2002","journal-title":"SIAM Rev."},{"key":"S012918312350105XBIB012","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","volume":"424","author":"Boccaletti S.","year":"2006","journal-title":"Phys. Rep."},{"key":"S012918312350105XBIB013","doi-asserted-by":"crossref","first-page":"036116","DOI":"10.1103\/PhysRevE.73.036116","volume":"73","author":"Karsai M.","year":"2006","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB014","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511791383","volume-title":"Dynamical Processes on Complex Networks","author":"Barrat A.","year":"2008"},{"key":"S012918312350105XBIB015","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511780356","volume-title":"Complex Networks: Structure, Robustness and Function","author":"Cohen R.","year":"2010"},{"key":"S012918312350105XBIB016","doi-asserted-by":"crossref","first-page":"068701","DOI":"10.1103\/PhysRevLett.111.068701","volume":"111","author":"Bogun\u00e1 M.","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB017","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1103\/RevModPhys.87.925","volume":"87","author":"Pastor-Satorras R.","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"S012918312350105XBIB018","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511524288","volume-title":"Nonequilibrium Phase Transitions in Lattice Models","author":"Marro J.","year":"1999"},{"key":"S012918312350105XBIB019","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1080\/00018730050198152","volume":"49","author":"Hinrichsen H.","year":"2000","journal-title":"Adv. Phys."},{"key":"S012918312350105XBIB020","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physa.2006.04.007","volume":"369","author":"Hinrichsen H.","year":"2006","journal-title":"Physica A"},{"key":"S012918312350105XBIB021","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1103\/RevModPhys.76.663","volume":"76","author":"\u00d3dor G.","year":"2004","journal-title":"Rev. Mod. Phys."},{"volume-title":"Non-Equilibrium Phase Transitions","year":"2008","author":"Henkel M.","key":"S012918312350105XBIB022"},{"key":"S012918312350105XBIB023","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1017\/S0305004100032680","volume":"53","author":"Broadbent S. R.","year":"1957","journal-title":"Proc. Camb. Philos. Soc."},{"key":"S012918312350105XBIB024","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1103\/RevModPhys.80.1275","volume":"80","author":"Dorogovtsev S. N.","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"S012918312350105XBIB025","doi-asserted-by":"crossref","first-page":"026117","DOI":"10.1103\/PhysRevE.86.026117","volume":"86","author":"\u00d3dor G.","year":"2012","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB026","doi-asserted-by":"crossref","first-page":"258701","DOI":"10.1103\/PhysRevLett.98.258701","volume":"98","author":"Hong H.","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB027","doi-asserted-by":"crossref","first-page":"062137","DOI":"10.1103\/PhysRevE.90.062137","volume":"90","author":"Wittmann M.","year":"2014","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB028","doi-asserted-by":"crossref","first-page":"066102","DOI":"10.1103\/PhysRevE.84.066102","volume":"84","author":"Ferreira S. C.","year":"2011","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB029","first-page":"358","volume":"2","author":"Lima F. W. S.","year":"2007","journal-title":"Commun. Comput. Phys."},{"key":"S012918312350105XBIB030","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1140\/epjb\/e2018-80551-9","volume":"91","author":"Krawiecki A.","year":"2018","journal-title":"Eur. Phys. J. B"},{"key":"S012918312350105XBIB031","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.12693\/APhysPolA.133.1433","volume":"133","author":"Krawiecki A.","year":"2018","journal-title":"Acta Phys. Pol. A"},{"key":"S012918312350105XBIB032","doi-asserted-by":"crossref","first-page":"91","DOI":"10.5506\/APhysPolBSupp.12.91","volume":"12","author":"Krawiecki A.","year":"2019","journal-title":"Acta Phys. Pol. B Proc. Suppl."},{"key":"S012918312350105XBIB033","doi-asserted-by":"crossref","first-page":"029802","DOI":"10.1103\/PhysRevLett.98.029802","volume":"98","author":"Castellano C.","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB034","doi-asserted-by":"crossref","first-page":"036110","DOI":"10.1103\/PhysRevE.79.036110","volume":"79","author":"Bogun\u00e1 M.","year":"2009","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB035","doi-asserted-by":"crossref","first-page":"027103","DOI":"10.1103\/PhysRevE.71.027103","volume":"71","author":"Catanzaro M.","year":"2005","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB036","doi-asserted-by":"crossref","first-page":"066113","DOI":"10.1103\/PhysRevE.83.066113","volume":"83","author":"Ferreira S. C.","year":"2011","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB037","doi-asserted-by":"crossref","first-page":"148701","DOI":"10.1103\/PhysRevLett.100.148701","volume":"100","author":"Castellano C.","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB038","doi-asserted-by":"crossref","first-page":"038701","DOI":"10.1103\/PhysRevLett.96.038701","volume":"96","author":"Castellano C.","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB039","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/0021-9991(76)90041-3","volume":"22","author":"Gillespie D. T.","year":"1976","journal-title":"J. Comput. Phys."},{"key":"S012918312350105XBIB040","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1021\/j100540a008","volume":"81","author":"Gillespie D. T.","year":"1977","journal-title":"J. Chem. Phys."},{"key":"S012918312350105XBIB041","doi-asserted-by":"crossref","first-page":"043208","DOI":"10.1088\/1742-5468\/aab04a","volume":"2018","author":"Macedo-Filho A.","year":"2018","journal-title":"J. Stat. Mech."},{"key":"S012918312350105XBIB042","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/j.physa.2018.07.013","volume":"510","author":"Mota M. P. S.","year":"2018","journal-title":"Physica A"},{"key":"S012918312350105XBIB043","doi-asserted-by":"crossref","first-page":"10205","DOI":"10.1088\/0305-4470\/35\/48\/301","volume":"35","author":"L\u00fcbeck S.","year":"2002","journal-title":"J. Phys. A, Math. Gen."},{"key":"S012918312350105XBIB044","doi-asserted-by":"crossref","first-page":"041126","DOI":"10.1103\/PhysRevE.76.041126","volume":"76","author":"Janssen H.-K.","year":"2007","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB045","first-page":"1","volume-title":"Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory","author":"Kenna R.","year":"2012"},{"key":"S012918312350105XBIB046","doi-asserted-by":"crossref","first-page":"115701","DOI":"10.1103\/PhysRevLett.96.115701","volume":"96","author":"Kenna R.","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB047","doi-asserted-by":"crossref","first-page":"155702","DOI":"10.1103\/PhysRevLett.97.155702","volume":"97","author":"Kenna R.","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"S012918312350105XBIB048","doi-asserted-by":"crossref","first-page":"011145","DOI":"10.1103\/PhysRevE.82.011145","volume":"82","author":"Palchykov V.","year":"2010","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB049","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1214\/aop\/1176996493","volume":"2","author":"Harris T. E.","year":"1974","journal-title":"Ann. Prob."},{"key":"S012918312350105XBIB050","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.physa.2005.05.051","volume":"357","author":"Dickman R.","year":"2005","journal-title":"Physica A"},{"key":"S012918312350105XBIB051","doi-asserted-by":"crossref","first-page":"016129","DOI":"10.1103\/PhysRevE.71.016129","volume":"71","author":"de Oliveira M. M.","year":"2005","journal-title":"Phys. Rev. E"},{"key":"S012918312350105XBIB052","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1214\/aoms\/1177706647","volume":"29","author":"Tukey J. W.","year":"1958","journal-title":"Ann. Math. Stat."},{"key":"S012918312350105XBIB053","doi-asserted-by":"crossref","first-page":"033203","DOI":"10.1088\/1742-5468\/ab75e7","volume":"2020","author":"Alves T. F. A.","year":"2020","journal-title":"J. Stat. Mech."}],"container-title":["International Journal of Modern Physics C"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S012918312350105X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,15]],"date-time":"2023-06-15T09:48:08Z","timestamp":1686822488000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S012918312350105X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,13]]},"references-count":53,"journal-issue":{"issue":"08","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["10.1142\/S012918312350105X"],"URL":"https:\/\/doi.org\/10.1142\/s012918312350105x","relation":{},"ISSN":["0129-1831","1793-6586"],"issn-type":[{"type":"print","value":"0129-1831"},{"type":"electronic","value":"1793-6586"}],"subject":[],"published":{"date-parts":[[2023,2,13]]},"article-number":"2350105"}}