{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,3]],"date-time":"2025-05-03T16:09:27Z","timestamp":1746288567366,"version":"3.37.3"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T00:00:00Z","timestamp":1662336000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T00:00:00Z","timestamp":1662336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100009873","name":"Regione Autonoma della Sardegna","doi-asserted-by":"publisher","award":["RASSR57257","RASSR57257"],"award-info":[{"award-number":["RASSR57257","RASSR57257"]}],"id":[{"id":"10.13039\/501100009873","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009112","name":"Istituto Nazionale di Alta Matematica \u201cFrancesco Severi\u201d","doi-asserted-by":"publisher","award":["Tecniche numeriche per l\u2019analisi delle reti complesse e lo studio dei problemi inversi","Tecniche numeriche per l\u2019analisi delle reti complesse e lo studio dei problemi inversi"],"award-info":[{"award-number":["Tecniche numeriche per l\u2019analisi delle reti complesse e lo studio dei problemi inversi","Tecniche numeriche per l\u2019analisi delle reti complesse e lo studio dei problemi inversi"]}],"id":[{"id":"10.13039\/100009112","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1720259"],"award-info":[{"award-number":["DMS-1720259"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Netw Sci"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The need to determine the structure of a graph arises in many applications. This paper studies directed graphs and defines the notions of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-chained and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{\\ell ,k\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-chained directed graphs. These notions reveal structural properties of directed graphs that shed light on how the nodes of the graph are connected. Applications include city planning, information transmission, and disease propagation. We also discuss the notion of in-center and out-center vertices of a directed graph, which are vertices at the center of the graph. Computed examples provide illustrations, among which is the investigation of a bus network for a city.<\/jats:p>","DOI":"10.1007\/s41109-022-00502-x","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T20:37:13Z","timestamp":1662410233000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Chained structure of directed graphs with applications to social and transportation networks"],"prefix":"10.1007","volume":"7","author":[{"given":"Anna","family":"Concas","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Caterina","family":"Fenu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Lothar","family":"Reichel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9054-8712","authenticated-orcid":false,"given":"Giuseppe","family":"Rodriguez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Yunzi","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,5]]},"reference":[{"key":"502_CR1","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.laa.2014.05.018","volume":"456","author":"J Baglama","year":"2014","unstructured":"Baglama J, Fenu C, Reichel L, Rodriguez G (2014) Analysis of directed networks via partial singular value decomposition and Gauss quadrature. Linear Algebra Appl 456:93\u2013121","journal-title":"Linear Algebra Appl"},{"key":"502_CR2","doi-asserted-by":"publisher","first-page":"2447","DOI":"10.1016\/j.laa.2012.10.022","volume":"438","author":"M Benzi","year":"2013","unstructured":"Benzi M, Estrada E, Klymko C (2013) Ranking hubs and authorities using matrix functions. Linear Algebra Appl 438:2447\u20132474","journal-title":"Linear Algebra Appl"},{"key":"502_CR3","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.socnet.2004.11.008","volume":"27","author":"SP Borgatti","year":"2005","unstructured":"Borgatti SP (2005) Centrality and network flow. Soc Netw 27:55\u201371","journal-title":"Soc Netw"},{"key":"502_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2019.06.022","volume":"373","author":"A Concas","year":"2020","unstructured":"Concas A, Noschese S, Reichel L, Rodriguez G (2020) A spectral method for bipartizing a network and detecting a large anti-community. J Comput Appl Math 373:112306","journal-title":"J Comput Appl Math"},{"key":"502_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/s41109-021-00377-4","volume":"6","author":"A Concas","year":"2021","unstructured":"Concas A, Reichel L, Rodriguez G, Zhang Y (2021) Chained graphs and some applications. Appl Netw Sci 6:39","journal-title":"Appl Netw Sci"},{"key":"502_CR6","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1007\/s11075-019-00704-y","volume":"83","author":"O De la Cruz Cabrera","year":"2020","unstructured":"De la Cruz Cabrera O, Matar M, Reichel L (2020) Edge importance in a network via line graphs and the matrix exponential. Numer Algorithms 83:807\u2013832","journal-title":"Numer Algorithms"},{"key":"502_CR7","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1007\/s11075-020-01050-0","volume":"88","author":"O De la Cruz Cabrera","year":"2021","unstructured":"De la Cruz Cabrera O, Matar M, Reichel L (2021) Centrality measures for node-weighted networks via line graphs and the matrix exponential. Numer Algorithms 88:583\u2013614","journal-title":"Numer Algorithms"},{"key":"502_CR8","volume-title":"Graph theory with applications to engineering and computer science","author":"N Deo","year":"1974","unstructured":"Deo N (1974) Graph theory with applications to engineering and computer science. Prentice-Hall, Englewood Cliffs"},{"key":"502_CR9","volume-title":"The structure of complex networks: theory and applications","author":"E Estrada","year":"2012","unstructured":"Estrada E (2012) The structure of complex networks: theory and applications. Oxford University Press, Oxford"},{"key":"502_CR10","doi-asserted-by":"publisher","first-page":"696","DOI":"10.1137\/090761070","volume":"52","author":"E Estrada","year":"2010","unstructured":"Estrada E, Higham DJ (2010) Network properties revealed through matrix functions. SIAM Rev 52:696\u2013714","journal-title":"SIAM Rev"},{"key":"502_CR11","volume-title":"A first course in network theory","author":"E Estrada","year":"2015","unstructured":"Estrada E, Knight P (2015) A first course in network theory. Oxford University Press, Oxford"},{"key":"502_CR12","volume":"71","author":"E Estrada","year":"2005","unstructured":"Estrada E, Rodriguez-Velazquez JA (2005) Subgraph centrality in complex networks. Phys Rev 71:056103","journal-title":"Phys Rev"},{"key":"502_CR13","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1016\/j.laa.2017.12.001","volume":"542","author":"D Fasino","year":"2017","unstructured":"Fasino D, Tudisco F (2017) A modularity based spectral method for simultaneous community and anti-community detection. Linear Algebra Appl 542:605\u2013623","journal-title":"Linear Algebra Appl"},{"key":"502_CR14","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1016\/j.physrep.2009.11.002","volume":"486","author":"S Fortunato","year":"2010","unstructured":"Fortunato S (2010) Community detection in graphs. Phys Rep 486:75\u2013174","journal-title":"Phys Rep"},{"key":"502_CR15","unstructured":"Free On-line Dictionary Of Computing. http:\/\/foldoc.org\/"},{"key":"502_CR16","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1137\/0207024","volume":"7","author":"HN Gabow","year":"1978","unstructured":"Gabow HN, Myers EW (1978) Finding all spanning trees of directed and undirected graphs. SIAM J Comput 7:280\u2013287","journal-title":"SIAM J Comput"},{"key":"502_CR17","unstructured":"Gephi Sample Data Sets. http:\/\/wiki.gephi.org\/index.php\/Datasets"},{"key":"502_CR18","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1145\/324133.324140","volume":"49","author":"J Kleinberg","year":"1999","unstructured":"Kleinberg J (1999) Authorative sources in hyperlinked environments. J ACM 49:604\u2013632","journal-title":"J ACM"},{"key":"502_CR19","unstructured":"Laenen S, Sun H (2020) Higher-order spectral clustering of directed graphs. In: Larochelle H, Ranzato M, Hadsell R, Balcan MF, Lin H (eds) Advances in neural information processing systems. Curran Associates, Inc., pp 941\u2013951. https:\/\/proceedings.neurips.cc\/paper\/2020\/file\/0a5052334511e344f15ae0bfafd47a67-Paper.pdf"},{"key":"502_CR20","unstructured":"Leskovec J, Krevl A (2014) SNAP Datasets: Stanford Large Network Dataset Collection. http:\/\/snap.stanford.edu\/data"},{"key":"502_CR21","unstructured":"MathOverflow website. https:\/\/mathoverflow.net\/"},{"key":"502_CR22","unstructured":"Matrix Market Collection. https:\/\/math.nist.gov\/MatrixMarket\/data\/Harwell-Boeing\/grenoble\/gre_1107.html"},{"key":"502_CR23","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199206650.001.0001","volume-title":"Networks: an introduction","author":"MEJ Newman","year":"2010","unstructured":"Newman MEJ (2010) Networks: an introduction. Oxford University Press, Oxford"},{"key":"502_CR24","volume-title":"The PageRank citation ranking: bringing order to the web","author":"L Page","year":"1999","unstructured":"Page L, Brin S, Motwani R, Winograd T (1999) The PageRank citation ranking: bringing order to the web. Stanford InfoLab, Stanford"},{"key":"502_CR25","unstructured":"Pajek dataset. http:\/\/vlado.fmf.uni-lj.si\/pub\/networks\/data\/"},{"key":"502_CR26","unstructured":"SuiteSparse Matrix Collection. https:\/\/sparse.tamu.edu\/HB\/ibm32\/"},{"key":"502_CR27","unstructured":"UF Sparse Matrix Collection. https:\/\/www.cise.ufl.edu\/research\/sparse\/matrices\/list_by_dimension.html"}],"container-title":["Applied Network Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41109-022-00502-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41109-022-00502-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41109-022-00502-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T20:47:21Z","timestamp":1662410841000},"score":1,"resource":{"primary":{"URL":"https:\/\/appliednetsci.springeropen.com\/articles\/10.1007\/s41109-022-00502-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,5]]},"references-count":27,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["502"],"URL":"https:\/\/doi.org\/10.1007\/s41109-022-00502-x","relation":{},"ISSN":["2364-8228"],"issn-type":[{"type":"electronic","value":"2364-8228"}],"subject":[],"published":{"date-parts":[[2022,9,5]]},"assertion":[{"value":"10 January 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 August 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 September 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}},{"value":"The authors declare that they have no competing interests.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"64"}}