{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T22:37:05Z","timestamp":1740177425138,"version":"3.37.3"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,12,1]],"date-time":"2019-12-01T00:00:00Z","timestamp":1575158400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,12,23]],"date-time":"2019-12-23T00:00:00Z","timestamp":1577059200000},"content-version":"vor","delay-in-days":22,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Netw Sci"],"published-print":{"date-parts":[[2019,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized configuration model, we partition the stubs in the configuration model into<jats:italic>b<\/jats:italic>blocks of equal sizes and choose a permutation function<jats:italic>h<\/jats:italic>for these blocks. In each block, we randomly designate a number proportional to<jats:italic>q<\/jats:italic>of stubs as type 1 stubs, where<jats:italic>q<\/jats:italic>is a parameter in the range [0,1]. Other stubs are designated as type 2 stubs. To construct a network, randomly select an unconnected stub. Suppose that this stub is in block<jats:italic>i<\/jats:italic>. If it is a type 1 stub, connect this stub to a randomly selected unconnected type 1 stub in block<jats:italic>h<\/jats:italic>(<jats:italic>i<\/jats:italic>). If it is a type 2 stub, connect it to a randomly selected unconnected type 2 stub. We repeat this process until all stubs are connected. Under an assumption, we derive a closed form for the joint degree distribution of two random neighboring vertices in the constructed graph. Based on this joint degree distribution, we show that the Pearson degree correlation function is linear in<jats:italic>q<\/jats:italic>for any fixed<jats:italic>b<\/jats:italic>. By properly choosing<jats:italic>h<\/jats:italic>, we show that our construction algorithm can create assortative networks as well as disassortative networks. We present a percolation analysis of this model. We verify our results by extensive computer simulations.<\/jats:p>","DOI":"10.1007\/s41109-019-0240-2","type":"journal-article","created":{"date-parts":[[2019,12,23]],"date-time":"2019-12-23T15:02:39Z","timestamp":1577113359000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A generalized configuration model with degree correlations and its percolation analysis"],"prefix":"10.1007","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4578-2002","authenticated-orcid":false,"given":"Duan-Shin","family":"Lee","sequence":"first","affiliation":[]},{"given":"Cheng-Shang","family":"Chang","sequence":"additional","affiliation":[]},{"given":"Miao","family":"Zhu","sequence":"additional","affiliation":[]},{"given":"Hung-Chih","family":"Li","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,23]]},"reference":[{"key":"240_CR1","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1126\/science.286.5439.509","volume":"286","author":"A-L Barab\u00e1si","year":"1999","unstructured":"Barab\u00e1si, A-L, Albert R (1999) Emergence of scaling in random networks. Science 286:509\u2013512.","journal-title":"Science"},{"key":"240_CR2","doi-asserted-by":"publisher","first-page":"083052","DOI":"10.1088\/1367-2630\/17\/8\/083052","volume":"17","author":"KE Bassler","year":"2015","unstructured":"Bassler, KE, Genio CID, Erdo\u030bs PL, Mikl\u00f3s I, Toroczkai Z (2015) Exact sampling of graphs with prescribed degree correlations. New J Phys 17:083052.","journal-title":"New J Phys"},{"key":"240_CR3","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/0097-3165(78)90059-6","volume":"24","author":"EA Bender","year":"1978","unstructured":"Bender, EA, Canfield ER (1978) The asymptotic number of labelled graphs with given degree sequences. J Comb Theory Ser A 24:296\u2013307.","journal-title":"J Comb Theory Ser A"},{"key":"240_CR4","doi-asserted-by":"publisher","first-page":"047104","DOI":"10.1103\/PhysRevE.66.047104","volume":"66","author":"M Bogu\u00f1\u00e1","year":"2002","unstructured":"Bogu\u00f1\u00e1, M, Pastor-Satorras R (2002) Epidemic spreading in correlated complex networks. Phys Rev E 66:047104.","journal-title":"Phys Rev E"},{"key":"240_CR5","doi-asserted-by":"publisher","first-page":"028701","DOI":"10.1103\/PhysRevLett.90.028701","volume":"90","author":"M Bogu\u00f1\u00e1","year":"2003","unstructured":"Bogu\u00f1\u00e1, M, Pastor-Satorras R, Vespignani A (2003) Absence of epidemic threshold in scale-free networks with degree correlations. Phys Rev Lett 90:028701.","journal-title":"Phys Rev Lett"},{"key":"240_CR6","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/S0195-6698(80)80030-8","volume":"1","author":"B Bollob\u00e1s","year":"1980","unstructured":"Bollob\u00e1s, B (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur J Comb 1:311\u2013316.","journal-title":"Eur J Comb"},{"key":"240_CR7","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/978-3-319-33781-4_8","volume-title":"Experimental Design Research","author":"Dan Braha","year":"2016","unstructured":"Braha, D (2016) The complexity of design networks: Structure and dynamics. In: Cash P, Mario TS, \u0160torga (eds)Experimental Design Research, 129\u2013151. https:\/\/doi.org\/10.1007\/978-3-319-33781-4_8."},{"issue":"7","key":"240_CR8","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1287\/mnsc.1060.0617","volume":"53","author":"D Braha","year":"2007","unstructured":"Braha, D, Bar-Yam Y (2007) The statistical mechanics of complex product development: Empirical and analytical results. Manag Sci 53(7):1127\u20131145.","journal-title":"Manag Sci"},{"key":"240_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8724-2","volume-title":"A Guide to Simulation, 2nd edn","author":"P Bratley","year":"1987","unstructured":"Bratley, P, Fox BL, Schrage LE (1987) A Guide to Simulation, 2nd edn. Springer, New York."},{"key":"240_CR10","doi-asserted-by":"publisher","first-page":"041902","DOI":"10.1103\/PhysRevE.64.041902","volume":"64","author":"DS Callaway","year":"2001","unstructured":"Callaway, DS, Hopcroft JE, Kleinberg JM, Newman MEJ, Strogatz SH (2001) Are randomly grown graphs really random?Phys Rev E 64:041902.","journal-title":"Phys Rev E"},{"issue":"6","key":"240_CR11","doi-asserted-by":"publisher","first-page":"L267","DOI":"10.1088\/0305-4470\/18\/6\/001","volume":"18","author":"JL Cardy","year":"1985","unstructured":"Cardy, JL, Grassberger P (1985) Epidemic models and percolation. J Phys A Math Gen 18(6):L267\u2013L271. https:\/\/doi.org\/10.1088\/0305-4470\/18\/6\/001. https:\/\/doi.org\/10.1088%2F0305-4470%2F18%2F6%2F001.","journal-title":"J Phys A Math Gen"},{"key":"240_CR12","doi-asserted-by":"publisher","first-page":"037101","DOI":"10.1103\/PhysRevE.70.037101","volume":"70","author":"M Catanzaro","year":"2004","unstructured":"Catanzaro, M, Caldarelli G, Pietronenero L (2004) Assortative model for social networks. Phys Rev E 70:037101.","journal-title":"Phys Rev E"},{"key":"240_CR13","doi-asserted-by":"publisher","first-page":"4626","DOI":"10.1103\/PhysRevLett.85.4626","volume":"85","author":"R Cohen","year":"2000","unstructured":"Cohen, R, Erez K, ben-Avraham D, Havlin S (2000) Resilence of the internet to random breakdowns. Phys Rev Lett 85:4626\u20134628.","journal-title":"Phys Rev Lett"},{"issue":"10, 108701","key":"240_CR14","doi-asserted-by":"publisher","first-page":"108701","DOI":"10.1103\/PhysRevLett.89.108701","volume":"89","author":"VM Egu\u00edluz","year":"2002","unstructured":"Egu\u00edluz, VM, Klemm K (2002) Epidemic threshold in structured scale-free networks. Phys Rev Lett 89(10, 108701):108701. https:\/\/doi.org\/10.1103\/PhysRevLett.89.108701.","journal-title":"Phys Rev Lett"},{"key":"240_CR15","first-page":"290","volume":"6","author":"Erdo\u030bs","year":"1959","unstructured":"Erdo\u030bs, R\u00e9nyi (1959) On random graphs. Publ Math 6:290\u2013297.","journal-title":"Publ Math"},{"key":"240_CR16","doi-asserted-by":"publisher","first-page":"1173","DOI":"10.1007\/s11117-018-0566-5","volume":"22","author":"C Guiver","year":"2018","unstructured":"Guiver, C (2018) On the strict monotonicity of spectral radii for classes of bounded positive linear operators. Positivity 22:1173\u20131190.","journal-title":"Positivity"},{"key":"240_CR17","doi-asserted-by":"publisher","first-page":"108702","DOI":"10.1103\/PhysRevLett.104.108702","volume":"104","author":"S Johnson","year":"2010","unstructured":"Johnson, S, Torres JJ, Marro J, Munoz MA (2010) The entropic origin of disassortativity in complex networks. Phys Rev Lett 104:108702.","journal-title":"Phys Rev Lett"},{"key":"240_CR18","doi-asserted-by":"publisher","first-page":"026101","DOI":"10.1103\/PhysRevE.85.026101","volume":"85","author":"H Klein-Hennig","year":"2012","unstructured":"Klein-Hennig, H, Hartmann AK (2012) Bias in generation of random graphs. Phys Rev E 85:026101.","journal-title":"Phys Rev E"},{"issue":"6","key":"240_CR19","first-page":"545","volume":"104","author":"W Kulpa","year":"1997","unstructured":"Kulpa, W (1997) The Poincar\u00e9-Miranda theorem. Am Math Mon 104(6):545\u2013550.","journal-title":"Am Math Mon"},{"key":"240_CR20","volume-title":"The Theory of Matrices","author":"P Lancaster","year":"1985","unstructured":"Lancaster, P, Tismenetsky M (1985) The Theory of Matrices. Academic Press, New York."},{"key":"240_CR21","volume-title":"Degree-degree correlations in random graphs with heavy-tailed degrees","author":"DN Litvak","year":"2012","unstructured":"Litvak, DN, van der Hofstad R (2012) Degree-degree correlations in random graphs with heavy-tailed degrees. Department of Applied Mathematics, University of Twente, Enschede, the Netherlands. http:\/\/doc.utwente.nl\/84367\/."},{"key":"240_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-68276-1","volume-title":"Inequalities: Theory of Majorization and Its Applications","author":"AW Marshall","year":"2011","unstructured":"Marshall, AW, Olkin I, Arnold BC (2011) Inequalities: Theory of Majorization and Its Applications. Springer, New York."},{"key":"240_CR23","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719512","volume-title":"Matrix Analysis and Applied Linear Algebra","author":"C Meyer","year":"2000","unstructured":"Meyer, C (2000) Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, USA."},{"key":"240_CR24","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1002\/rsa.3240060204","volume":"6","author":"M Molloy","year":"1995","unstructured":"Molloy, M, Reed B (1995) A critical point for random graphs with a given degree sequence. Random Struct Alg 6:161\u2013179.","journal-title":"Random Struct Alg"},{"key":"240_CR25","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1017\/S0963548398003526","volume":"7","author":"M Molloy","year":"1998","unstructured":"Molloy, M, Reed B (1998) The size of the giant component of a random graph with a given degree sequence. Comb Probab Comput 7:295\u2013306.","journal-title":"Comb Probab Comput"},{"key":"240_CR26","doi-asserted-by":"publisher","first-page":"5678","DOI":"10.1103\/PhysRevE.61.5678","volume":"61","author":"C Moore","year":"2000","unstructured":"Moore, C, Newman MEJ (2000) Epidemics and percolation in small-world networks. Phys Rev E 61:5678.","journal-title":"Phys Rev E"},{"key":"240_CR27","doi-asserted-by":"publisher","first-page":"035103","DOI":"10.1103\/PhysRevE.68.035103","volume":"68","author":"Y Moreno","year":"2003","unstructured":"Moreno, Y, G\u00f3mez JB, Pacheco AF (2003) Epidemic incidence in correlated complex networks. Phys Rev E 68:035103.","journal-title":"Phys Rev E"},{"key":"240_CR28","doi-asserted-by":"publisher","first-page":"025102","DOI":"10.1103\/PhysRevE.64.025102","volume":"64","author":"MEJ Newman","year":"2001","unstructured":"Newman, MEJ (2001) Clustering and preferential attachment in growing networks. Phys Rev E 64:025102.","journal-title":"Phys Rev E"},{"key":"240_CR29","doi-asserted-by":"publisher","first-page":"208701","DOI":"10.1103\/PhysRevLett.89.208701","volume":"89","author":"MEJ Newman","year":"2002","unstructured":"Newman, MEJ (2002) Assortative mixing in networks. Phys Rev Lett 89:208701.","journal-title":"Phys Rev Lett"},{"key":"240_CR30","doi-asserted-by":"publisher","first-page":"026126","DOI":"10.1103\/PhysRevE.67.026126","volume":"67","author":"MEJ Newman","year":"2003","unstructured":"Newman, MEJ (2003) Mixing patterns on networks. Phys Rev E 67:026126.","journal-title":"Phys Rev E"},{"key":"240_CR31","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199206650.001.0001","volume-title":"Networks: An Introduction","author":"M Newman","year":"2010","unstructured":"Newman, M (2010) Networks: An Introduction. Oxford University Press, New York."},{"key":"240_CR32","doi-asserted-by":"crossref","unstructured":"Nikoloski, Z, Deo N, Kucera L (2005) Degree-correlation of a scale-free random graph process. In: Stefan F (ed)2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb \u201905), 239\u2013244. http:\/\/www.dmtcs.org\/proceedings\/html\/dmAE0148.abs.html.","DOI":"10.46298\/dmtcs.3406"},{"key":"240_CR33","doi-asserted-by":"publisher","first-page":"8209","DOI":"10.1073\/pnas.0900142106","volume":"106","author":"A Pomerance","year":"2009","unstructured":"Pomerance, A, Ott E, Girvan M, Losert W (2009) The effect of network topology on the stability of discrete state models of genetic control. Proc Natl Acad Sci 106:8209\u20138214.","journal-title":"Proc Natl Acad Sci"},{"key":"240_CR34","doi-asserted-by":"publisher","first-page":"046107","DOI":"10.1103\/PhysRevE.67.046107","volume":"67","author":"A Ramezanpour","year":"2005","unstructured":"Ramezanpour, A, Karimipour V, Mashaghi A (2005) Generating correlated networks from uncorrelated ones. Phys Rev E 67:046107.","journal-title":"Phys Rev E"},{"key":"240_CR35","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1090\/S0025-5718-1973-0325650-4","volume":"27","author":"WC Rheinboldt","year":"1973","unstructured":"Rheinboldt, WC, Vandergraft JS (1973) A simple approach to the perron\u2013frobenius theory for positive operators on general partially-ordered finite-dimensional linear spaces. Math Comput 27:139\u2013145.","journal-title":"Math Comput"},{"key":"240_CR36","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1016\/S0025-5564(02)00117-7","volume":"80","author":"LM Sander","year":"2002","unstructured":"Sander, LM, Warren CP, Sokolov IM, Simon C, Koopman J (2002) Percolation on heterogeneous networks as a model for epidemics. Math Biosci 80:293\u2013305.","journal-title":"Math Biosci"},{"key":"240_CR37","doi-asserted-by":"publisher","first-page":"015104","DOI":"10.1103\/PhysRevE.66.015104","volume":"66","author":"N Schwartz","year":"2002","unstructured":"Schwartz, N, Cohen R, ben-Avraham D, Barabasi A-L, Havlin S (2002) Percolation in directed scale-free networks. Phys Rev E 66:015104.","journal-title":"Phys Rev E"},{"key":"240_CR38","doi-asserted-by":"publisher","first-page":"015101","DOI":"10.1103\/PhysRevE.85.015101","volume":"85","author":"M Schl\u00e4pfer","year":"2012","unstructured":"Schl\u00e4pfer, M, Buzna L (2012) Decelerated spreading in degree-correlated networks. Phys Rev E 85:015101.","journal-title":"Phys Rev E"},{"key":"240_CR39","doi-asserted-by":"publisher","first-page":"015101","DOI":"10.1103\/PhysRevE.67.015101","volume":"67","author":"A V\u00e1zquez","year":"2003","unstructured":"V\u00e1zquez, A, Moreno Y (2003) Resilence to damage of graphs with degree correlations. Phys Rev E 67:015101.","journal-title":"Phys Rev E"},{"key":"240_CR40","doi-asserted-by":"publisher","first-page":"440","DOI":"10.1038\/30918","volume":"393","author":"DJ Watts","year":"1998","unstructured":"Watts, DJ, Strogatz SH (1998) Collective dynamics of \u2019small-world\u2019 networks. Nature 393:440\u2013442.","journal-title":"Nature"},{"issue":"5","key":"240_CR41","first-page":"1431","volume":"36","author":"R Xulvi-Brunet","year":"2005","unstructured":"Xulvi-Brunet, R, Sokolov IM (2005) Changing correlations in networks: assortativity and dissortativity. Acta Phys Pol B 36(5):1431\u20131455.","journal-title":"Acta Phys Pol B"},{"issue":"11","key":"240_CR42","doi-asserted-by":"publisher","first-page":"3495","DOI":"10.1142\/S0218127408022536","volume":"18","author":"J Zhou","year":"2008","unstructured":"Zhou, J, Xu X, Zhang J, Sun J, Small M, Lu J-A (2008) Generating an assortative network with a given degree distribution. Intern J Bifuration Chaos 18(11):3495\u20133502.","journal-title":"Intern J Bifuration Chaos"}],"container-title":["Applied Network Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s41109-019-0240-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s41109-019-0240-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s41109-019-0240-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,9]],"date-time":"2022-10-09T07:28:49Z","timestamp":1665300529000},"score":1,"resource":{"primary":{"URL":"https:\/\/appliednetsci.springeropen.com\/articles\/10.1007\/s41109-019-0240-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12]]},"references-count":42,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["240"],"URL":"https:\/\/doi.org\/10.1007\/s41109-019-0240-2","relation":{},"ISSN":["2364-8228"],"issn-type":[{"type":"electronic","value":"2364-8228"}],"subject":[],"published":{"date-parts":[[2019,12]]},"assertion":[{"value":"12 August 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 November 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that we have no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"124"}}