{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,3]],"date-time":"2025-12-03T17:41:54Z","timestamp":1764783714626,"version":"3.40.5"},"reference-count":21,"publisher":"Wiley","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:p>Calculation of assortative mixing by degree in networks indicates whether nodes with similar degree are connected to each other. In networks with scale-free distribution high values of assortative mixing by degree can be an indication of a hub-like core in networks. Degree correlation has generally been used to measure assortative mixing of a network. But it has been shown that degree correlation cannot always distinguish properly between different networks with nodes that have the same degrees. The so-called<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>-metric has been shown to be a better choice to calculate assortative mixing. The<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>-metric is normalized with respect to the class of networks without self-loops, multiple edges, and multiple components, while degree correlation is always normalized with respect to unrestricted networks, where self-loops, multiple edges, and multiple components are allowed. The challenge in computing the normalized<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>-metric is in obtaining the minimum and maximum value within a specific class of networks. We show that this can be solved by using linear programming. We use Lagrangian relaxation and the subgradient algorithm to obtain a solution to the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>-metric problem. Several examples are given to illustrate the principles and some simulations indicate that the solutions are generally accurate.<\/jats:p>","DOI":"10.1155\/2015\/580361","type":"journal-article","created":{"date-parts":[[2015,2,22]],"date-time":"2015-02-22T16:01:06Z","timestamp":1424620866000},"page":"1-9","source":"Crossref","is-referenced-by-count":1,"title":["Computing Assortative Mixing by Degree with the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>s<\/mml:mi><\/mml:mrow><\/mml:math>-Metric in Networks Using Linear Programming"],"prefix":"10.1155","volume":"2015","author":[{"given":"Lourens J.","family":"Waldorp","sequence":"first","affiliation":[{"name":"University of Amsterdam, Weesperplein 4, 1018 XA Amsterdam, Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Verena D.","family":"Schmittmann","sequence":"additional","affiliation":[{"name":"Tilburg University, Warandelaan 2, 5037 AB Tilburg, Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"issue":"20","key":"19","volume":"89","year":"2002","journal-title":"Physical Review Letters"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2010-00219-x"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1126\/science.286.5439.509"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1017\/cbo9780511791383"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0027407"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1080\/15427951.2005.10129111"},{"key":"1","doi-asserted-by":"publisher","DOI":"10.1103\/physreve.75.046102"},{"year":"1986","series-title":"Wiley-Interscience Series in Discrete Mathematics","key":"24"},{"year":"1999","key":"18"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1287\/opre.18.6.1138"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1140\/epjb\/e2009-00219-y"},{"year":"1997","key":"5"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218x(86)90056-9"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.1040.0263"},{"first-page":"33","volume-title":"Structure of the extreme points of the subtour elimination problem of the stsp","year":"2010","key":"8"},{"year":"1999","series-title":"Wiley Series in Probability and Statistics","key":"17"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01580223"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/bf01584070"},{"year":"2012","key":"22"},{"year":"2004","key":"4"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1017\/s0140525x09991567"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/580361.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/580361.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2015\/580361.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,27]],"date-time":"2016-07-27T16:36:09Z","timestamp":1469637369000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2015\/580361\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"references-count":21,"alternative-id":["580361","580361"],"URL":"https:\/\/doi.org\/10.1155\/2015\/580361","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2015]]}}}