{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T18:38:28Z","timestamp":1791916708085,"version":"4.3.4"},"reference-count":72,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, communicability, and betweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network's adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks.<\/jats:p>","DOI":"10.1137\/090761070","type":"journal-article","created":{"date-parts":[[2010,11,8]],"date-time":"2010-11-08T18:17:23Z","timestamp":1289240243000},"page":"696-714","source":"Crossref","is-referenced-by-count":256,"title":["Network Properties Revealed through Matrix Functions"],"prefix":"10.1137","volume":"52","author":[{"given":"Ernesto","family":"Estrada","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Desmond J.","family":"Higham","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,8]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.74.47"},{"key":"R2","unstructured":"U. Alon,\n                      An Introduction to Systems Biology\n                      , Chapman & Hall\/CRC, London, 2006."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1126\/science.286.5439.509"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2004.10.015"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1680020402"},{"key":"R6","doi-asserted-by":"crossref","first-page":"16","DOI":"10.17730\/humo.7.3.f4033344851gl053","volume":"7","author":"Bavelas A.","year":"1948","journal-title":"Human Organization","ISSN":"https:\/\/id.crossref.org\/issn\/0018-7259","issn-type":"print"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1121\/1.1906679"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2005.10.009"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1080\/0022250X.1972.9989806"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1086\/228631"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.socnet.2005.11.005"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1093\/nar\/gkg340"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/s10910-007-9314-y"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0510525103"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"D. P. Croft, R. James, and J. Krause,\n                      Exploring Animal Social Networks\n                      , Princeton University Press, Princeton, NJ, 2008.","DOI":"10.1515\/9781400837762"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1098\/rsif.2008.0484"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"D. Cvetkov\u0107, P. Rowlinson, and S. Simi\u0107,\n                      Eigenspaces of Graphs\n                      , Cambridge University Press, Cambridge, UK, 1997.","DOI":"10.1017\/CBO9781139086547"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1080\/00018730601170527"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1126\/science.1081058"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1080\/0022250X.1974.9989837"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"L. Eld\u00e9n,\n                      Matrix Methods in Data Mining and Pattern Recognition\n                      , SIAM, Philadelphia, PA, 2007.","DOI":"10.1137\/1.9780898718867"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/S0009-2614(00)00158-5"},{"key":"R23","first-page":"563","volume":"52","author":"Estrada E.","year":"2006","journal-title":"Eur. Phys. J. B Condens. Matter Phys."},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1209\/epl\/i2005-10441-3"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1002\/pmic.200500209"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.ecocom.2007.02.018"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1890\/07-1419.1"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.77.036111"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2007.03.098"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2008.11.011"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.71.056103"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1016\/0378-8733(78)90021-7"},{"key":"R33","unstructured":"J. E. Gentle,\n                      Elements of Computational Statistics\n                      , Springer, New York, 2002."},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2008.02.026"},{"key":"R35","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996."},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2009.0456"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1586\/14789450.1.2.229"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2007.01.044"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.2298\/JSC0712321G"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2007.08.053"},{"key":"R41","doi-asserted-by":"crossref","unstructured":"B. W. Herr, W. Ke, E. Hardy, and K. B\u00f6rner,\n                      Movies and actors: Mapping the Internet Movie Database\n                      , in Proceedings of the 11th International Conference on Information Visualization, 2007, pp. 465\u2013469.","DOI":"10.1109\/IV.2007.78"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2006.04.026"},{"key":"R43","doi-asserted-by":"crossref","unstructured":"N. J. Higham,\n                      Functions of Matrices: Theory and Computation\n                      , SIAM, Philadelphia, PA, 2008.","DOI":"10.1137\/1.9780898717778"},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.2307\/2785990"},{"key":"R45","unstructured":"T. P. Hughes,\n                      Networks of Power: Electrification in Western Society, 1880\u20131930\n                      , Johns Hopkins University Press, Baltimore, MD, 1993."},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.0020095"},{"key":"R47","doi-asserted-by":"crossref","unstructured":"L. Kamper, A. Bozkurt, K. Rybacki, A. Geissler, I. Gerken, K. E. Stephan, and R. K\u00f6tter,\n                      An introduction to CoCoMac-Online. The online-interface of the primate connectivity database CoCoMac\n                      , in Neuroscience Databases: A Practical Guide, R. K\u00f6tter, ed., Kluwer Academic, Norwell, MA, 2002, pp. 155\u2013169.","DOI":"10.1007\/978-1-4615-1079-6_11"},{"key":"R48","unstructured":"B. Kapferer,\n                      Norms and the manipulation of relationships in a work context\n                      , in Social Networks in Urban Situations, Manchester University Press, Manchester, UK, 1969."},{"key":"R49","unstructured":"B. Kapferer,\n                      Strategy and Transaction in an African Factory\n                      , Manchester University Press, Manchester, UK, 1972."},{"key":"R50","doi-asserted-by":"publisher","DOI":"10.1007\/BF02289026"},{"key":"R51","doi-asserted-by":"publisher","DOI":"10.1038\/35022643"},{"key":"R52","doi-asserted-by":"publisher","DOI":"10.1385\/NI:2:2:127"},{"key":"R53","doi-asserted-by":"crossref","unstructured":"A. N. Langville and C. D. Meyer,\n                      Google's PageRank and Beyond: The Science of Search Engine Rankings\n                      , Princeton University Press, Princeton, NJ, 2006.","DOI":"10.1515\/9781400830329"},{"key":"R54","doi-asserted-by":"publisher","DOI":"10.1037\/h0057189"},{"key":"R55","doi-asserted-by":"publisher","DOI":"10.3354\/meps230001"},{"key":"R56","unstructured":"D. MacKay,\n                      Information Theory, Inference, and Learning Algorithms\n                      , Cambridge University Press, Cambridge, UK, 2003."},{"key":"R57","first-page":"159","volume":"16","author":"Manke T.","year":"2005","journal-title":"Genome Informatics"},{"key":"R58","first-page":"60","volume":"2","author":"Milgram S.","year":"1967","journal-title":"Psychol. Today","ISSN":"https:\/\/id.crossref.org\/issn\/0033-3107","issn-type":"print"},{"key":"R59","doi-asserted-by":"crossref","unstructured":"J. L. Morrison, R. Breitling, D. J. Higham, and D. R. Gilbert,\n                      Generank: Using search engine technology for the analysis of microarray experiments\n                      , BMC Bioinformatics, 6 (233) (2005); available online from http:\/\/www.biomedcentral.com\/1471-2105\/6\/233.","DOI":"10.1186\/1471-2105-6-233"},{"key":"R60","doi-asserted-by":"publisher","DOI":"10.1093\/bioinformatics\/btl338"},{"key":"R61","first-page":"44","volume":"147","author":"Newman M. E. J.","year":"2001","journal-title":"Comput. Phys. Comm.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-4655","issn-type":"print"},{"key":"R62","doi-asserted-by":"publisher","DOI":"10.1016\/j.socnet.2004.11.009"},{"key":"R63","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0610245104"},{"key":"R64","unstructured":"L. Page, S. Brin, R. Motwani, and T. Winograd,\n                      The PageRank Citation Ranking: Bringing Order to the Web\n                      , Tech. report, Stanford Digital Library Technologies Project, 1998."},{"key":"R65","doi-asserted-by":"publisher","DOI":"10.1385\/NI:2:2:145"},{"key":"R66","unstructured":"G. Strang,\n                      Computational Science and Engineering\n                      , Wellesly-Cambridge Press, Cambridge, MA, 2007."},{"key":"R67","doi-asserted-by":"publisher","DOI":"10.1145\/1462173.1462175"},{"key":"R68","doi-asserted-by":"crossref","unstructured":"L. N. Trefethen and M. Embree,\n                      Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators\n                      , Princeton University Press, Princeton, NJ, 2005.","DOI":"10.1515\/9780691213101"},{"key":"R69","doi-asserted-by":"publisher","DOI":"10.1038\/nature750"},{"key":"R70","unstructured":"S. Wassermann and K. Faust,\n                      Social Network Analysis: Methods and Applications\n                      , Cambridge University Press, Cambridge, UK, 1994."},{"key":"R71","doi-asserted-by":"publisher","DOI":"10.1038\/30918"},{"key":"R72","unstructured":"P. Whoriskey,\n                      Instant-messagers really are about six degrees from Kevin Bacon: Big Microsoft study supports small world theory\n                      , Washington Post, August 2, 2008; available online from http:\/\/www.washingtonpost.com\/wp-dyn\/content\/article\/2008\/08\/01\/AR2008080103718.html."}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090761070","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:45:02Z","timestamp":1787229902000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090761070"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":72,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/090761070"],"URL":"https:\/\/doi.org\/10.1137\/090761070","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}