{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T11:46:41Z","timestamp":1753876001456,"version":"3.41.2"},"reference-count":57,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2020,6,1]],"date-time":"2020-06-01T00:00:00Z","timestamp":1590969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this article, we present a novel approach to analyse the structure of complex networks represented by a quantum graph. A quantum graph is a metric graph with a differential operator (including the edge-based Laplacian) acting on functions defined on the edges of the graph. Every edge of the graph has a length interval assigned to it. The structural information contents are measured using graph entropy which has been proved useful to analyse and compare the structure of complex networks. Our definition of graph entropy is based on local edge functionals. These edge functionals are obtained by a diffusion process defined using the edge-based Laplacian of the graph using the quantum graph representation. We first present the general framework to define graph entropy using heat diffusion process and discuss some of its properties for different types of network models. Second, we propose a novel signature to gauge the structural complexity of the network and apply the proposed method to different datasets.<\/jats:p>","DOI":"10.1093\/comnet\/cnaa015","type":"journal-article","created":{"date-parts":[[2020,5,31]],"date-time":"2020-05-31T03:26:47Z","timestamp":1590895607000},"source":"Crossref","is-referenced-by-count":4,"title":["Network entropy using edge-based information functionals"],"prefix":"10.1093","volume":"8","author":[{"given":"Furqan","family":"Aziz","sequence":"first","affiliation":[{"name":"Centre for Computational Biology, University of Birmingham, Birmingham B15 2TT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Edwin R","family":"Hancock","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of York, York YO10 5GH, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richard C","family":"Wilson","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of York, York YO10 5GH, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2020,8,25]]},"reference":[{"key":"2020082514270474900_B1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198805090.001.0001","volume-title":"Networks","author":"Newman,","year":"2018"},{"key":"2020082514270474900_B2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1371\/journal.pcbi.1005466","article-title":"A review of active learning approaches to experimental design for uncovering biological networks","volume":"13","author":"Sverchkov,","year":"2017","journal-title":"PLOS Comput. Biol."},{"key":"2020082514270474900_B3","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.ijinfomgt.2017.08.003","article-title":"Social network analysis: characteristics of online social networks after a disaster","volume":"38","author":"Kim,","year":"2018","journal-title":"Int. J. Inf. Manag."},{"key":"2020082514270474900_B4","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1093\/comnet\/cnx023","article-title":"Representation of texts as complex networks: a mesoscopic approach","volume":"6","author":"Ferraz de Arruda,","year":"2017","journal-title":"J. Compl. Netw."},{"key":"2020082514270474900_B5","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1093\/comnet\/cnt005","article-title":"Predicting efficiency in master\u2013slave grid computing systems","volume":"1","author":"Travieso,","year":"2013","journal-title":"J. Compl. Netw."},{"key":"2020082514270474900_B6","first-page":"041036","article-title":"Inheritance patterns in citation networks reveal scientific memes","volume":"4","author":"Kuhn,","year":"2014","journal-title":"Phys. Rev. X"},{"key":"2020082514270474900_B7","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.patcog.2017.07.025","article-title":"A hybrid reproducing graph kernel based on information entropy","volume":"73","author":"Xu,","year":"2018","journal-title":"Patt. Recogn."},{"key":"2020082514270474900_B8","doi-asserted-by":"crossref","first-page":"045102","DOI":"10.1103\/PhysRevE.80.045102","article-title":"Entropy measures for networks: toward an information theory of complex topologies","volume":"80","author":"Anand,","year":"2009","journal-title":"Phys. Rev. E"},{"key":"2020082514270474900_B9","article-title":"Coding of an information source having ambiguous alphabet and the entropy of graphs","volume-title":"6th Prague Conference on Information Theory","author":"K\u00f6rner,","year":"1973"},{"key":"2020082514270474900_B10","first-page":"82","article-title":"Information processing in complex networks: graph entropy and information functionals","volume":"201","author":"Dehmer,","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"2020082514270474900_B11","doi-asserted-by":"crossref","first-page":"58","DOI":"10.4018\/jats.2009071005","article-title":"Quantifying complexity in networks: The von Neumann entropy","volume":"1","author":"Passerini,","year":"2009","journal-title":"Int J Agent Technol Syst."},{"key":"2020082514270474900_B12","first-page":"1958","article-title":"Graph characterizations from von Neumann entropy","volume":"11","author":"Han,","year":"2102","journal-title":"Pattern Recogn. Lett."},{"key":"2020082514270474900_B13","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.laa.2005.09.008","article-title":"Laplacian energy of a graph","volume":"414","author":"Gutman,","year":"2006","journal-title":"Linear Algebra Appl."},{"key":"2020082514270474900_B14","doi-asserted-by":"crossref","first-page":"066102","DOI":"10.1103\/PhysRevE.82.066102","article-title":"Quantifying network heterogeneity","volume":"82","author":"Estrada,","year":"2010","journal-title":"Phys. Rev. E"},{"key":"2020082514270474900_B15","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1016\/S0009-2614(00)00158-5","article-title":"Characterization of 3D molecular structure","volume":"319","author":"Estrada,","year":"2000","journal-title":"Chem. Phys. Lett."},{"key":"2020082514270474900_B16","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1093\/bioinformatics\/18.5.697","article-title":"Characterization of the folding degree of proteins","volume":"18","author":"Estrada,","year":"2002","journal-title":"Bioinformatics"},{"key":"2020082514270474900_B17","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1007\/978-3-319-49055-7_16","volume-title":"Structural, Syntactic, and Statistical Pattern Recognition","author":"Aziz,","year":"2016"},{"key":"2020082514270474900_B18","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1080\/01969720802435925","article-title":"A novel method for measuring the structural information content of networks","volume":"39","author":"Dehmer,","year":"2008","journal-title":"Cybern. Syst."},{"key":"2020082514270474900_B19","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.laa.2013.11.009","article-title":"Walk entropies in graphs","volume":"443","author":"Estrada,","year":"2014","journal-title":"Linear Algebra Appl."},{"key":"2020082514270474900_B20","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1109\/TNNLS.2013.2248093","article-title":"Backtrackless walks on a graph","volume":"24","author":"Aziz,","year":"2013","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"2020082514270474900_B21","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1016\/j.patcog.2017.12.003","article-title":"Product graph-based higher order contextual similarities for inexact subgraph matching","volume":"76","author":"Dutta,","year":"2018","journal-title":"Pattern Recogn"},{"key":"2020082514270474900_B22","first-page":"3822","article-title":"Shape signature using the edge-based Laplacian","volume-title":"International Conference on Pattern Recognition","author":"Aziz,","year":"2012"},{"key":"2020082514270474900_B23","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1093\/comnet\/cny023","article-title":"A wave packet signature for complex networks","volume":"7","author":"Aziz,","year":"2018","journal-title":"J. Compl. Netw."},{"key":"2020082514270474900_B24","doi-asserted-by":"crossref","first-page":"2347","DOI":"10.1145\/3219819.3219991","article-title":"NetLSD: hearing the shape of a graph","volume-title":"Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery &#38; Data Mining","author":"Tsitsulin,","year":"2018"},{"key":"2020082514270474900_B25","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1109\/TNN.2010.2091969","article-title":"Graph characterization via Ihara coefficients","volume":"22","author":"Ren,","year":"2011","journal-title":"IEEE Trans. Neural Netw."},{"key":"2020082514270474900_B26","article-title":"Calculus on graphs","author":"Friedman,","year":"2004","journal-title":"CoRR"},{"key":"2020082514270474900_B27","first-page":"291","volume-title":"Proceedings of Symposium on Pure Mathematics","author":"Kuchment,","year":"2008"},{"key":"2020082514270474900_B28","doi-asserted-by":"crossref","first-page":"229","DOI":"10.2140\/pjm.2004.216.229","article-title":"Wave equations for graphs and the edge based Laplacian","volume":"216","author":"Friedman,","year":"2004","journal-title":"Pac. J. Math."},{"key":"2020082514270474900_B29","doi-asserted-by":"crossref","first-page":"4183","DOI":"10.1016\/j.laa.2013.01.007","article-title":"Eigenfunctions of the edge-based Laplacian on a graph","volume":"438","author":"Wilson,","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"2020082514270474900_B30","first-page":"1383","article-title":"A concise and provably informative multi-scale signature based on heat diffusion","author":"Sun,","year":"2009","journal-title":"Proceedings of the Symposium on Geometry Processing"},{"key":"2020082514270474900_B31","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.cplett.2007.03.098","article-title":"Statistical-mechanical approach to subgraph centrality in complex networks","volume":"439","author":"Estrada,","year":"2007","journal-title":"Chem. Phys. Lett."},{"key":"2020082514270474900_B32","first-page":"17","article-title":"On the evolution of random graphs","volume":"286","author":"Erd\u00f5s,","year":"1960","journal-title":"Public. Math. Inst. Hung. Acad. Sci."},{"key":"2020082514270474900_B33","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","author":"Watts,","year":"1998","journal-title":"Nature"},{"key":"2020082514270474900_B34","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Barab\u00e1si,","year":"1999","journal-title":"Science"},{"key":"2020082514270474900_B35","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0375-9601(99)00757-4","article-title":"Renormalization group analysis of the small-world network model","volume":"263","author":"Newman,","year":"1999","journal-title":"Phys. Lett. A"},{"key":"2020082514270474900_B36","doi-asserted-by":"crossref","first-page":"026107","DOI":"10.1103\/PhysRevE.65.026107","article-title":"Growing scale-free networks with tunable clustering","volume":"65","author":"Holme,","year":"2002","journal-title":"Phys. Rev. E"},{"key":"2020082514270474900_B37","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1017\/S0963548399003867","article-title":"Generating random regular graphs quickly","volume":"8","author":"Steger,","year":"1999","journal-title":"Combin., Probab. Comput."},{"key":"2020082514270474900_B38","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1109\/ICDM.2016.0142","article-title":"Faster kernels for graphs with continuous attributes via hashing","author":"Morris,","year":"2016","journal-title":"2016 IEEE 16th International Conference on Data Mining (ICDM)"},{"key":"2020082514270474900_B39","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1145\/2487788.2488173","article-title":"KONECT\u2014the Koblenz network collection","volume-title":"Proceedings of the International Conference on World Wide Web Companion","author":"Kunegis,","year":"2013"},{"key":"2020082514270474900_B40","doi-asserted-by":"crossref","first-page":"858","DOI":"10.1093\/comnet\/cnx017","article-title":"Spin statistics, partition functions and network entropy","volume":"5","author":"Wang,","year":"2017","journal-title":"J. Compl. Netw."},{"key":"2020082514270474900_B41","doi-asserted-by":"crossref","first-page":"032810","DOI":"10.1103\/PhysRevE.92.032810","article-title":"Thermodynamic characterization of networks using graph polynomials","volume":"92","author":"Ye,","year":"2015","journal-title":"Phys. Rev. E"},{"key":"2020082514270474900_B42","article-title":"Columbia object image library (coil 100)","volume-title":"Department of Computer Science","author":"Nayar,","year":"1996"},{"key":"2020082514270474900_B43","first-page":"147","article-title":"A combined corner and edge detector","volume-title":"Fourth Alvey Vision Conference","author":"Harris,","year":"1988"},{"key":"2020082514270474900_B44","first-page":"793","article-title":"Sur la sph\u00e9re vide","volume":"6","author":"Delaunay,","year":"1934","journal-title":"Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk."},{"key":"2020082514270474900_B45","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0004-3702(95)00122-0","article-title":"Theories for mutagenicity: a study in first-order and feature-based induction","volume":"85","author":"Srinivasan,","year":"1996","journal-title":"Artif. Intell."},{"key":"2020082514270474900_B46","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1093\/bioinformatics\/17.1.107","article-title":"The predictive toxicology challenge 2000\u20132001","volume":"17","author":"Helma,","year":"2001","journal-title":"Bioinformatics"},{"key":"2020082514270474900_B47","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/978-3-540-89689-0_33","article-title":"IAM graph database repository for graph based pattern recognition and machine learning","volume-title":"Structural, Syntactic, and Statistical Pattern Recognition","author":"Riesen,","year":"2008"},{"key":"2020082514270474900_B48","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/s10115-007-0103-5","article-title":"Comparison of descriptor spaces for chemical compound retrieval and classification","volume":"14","author":"Wale,","year":"2008","journal-title":"Knowl. Inf. Syst."},{"key":"2020082514270474900_B49","first-page":"3756","article-title":"Graph invariant kernels","volume-title":"Proceedings of the 24th International Conference on Artificial Intelligence, IJCAI\u201915","author":"Orsini,","year":"2015"},{"key":"2020082514270474900_B50","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1021\/jm040835a","article-title":"Derivation and validation of toxicophores for mutagenicity prediction","volume":"48","author":"Kazius,","year":"2005","journal-title":"J. Med. Chem."},{"key":"2020082514270474900_B51","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1007\/s41109-019-0195-3","article-title":"A survey on graph kernels","volume":"5","author":"Kriege,","year":"2020","journal-title":"Appl. Netw. Sci."},{"key":"2020082514270474900_B52","doi-asserted-by":"crossref","first-page":"3508","DOI":"10.1093\/bioinformatics\/bth436","article-title":"Modeling interactome: scale-free or geometric?","volume":"20","author":"Pr\u017eulj,","year":"2004","journal-title":"Bioinformatics"},{"key":"2020082514270474900_B53","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/978-3-540-45167-9_11","article-title":"On graph kernels: hardness results and efficient alternatives","volume-title":"Learning Theory and Kernel Machines","author":"G\u00e4rtner,","year":"2003"},{"key":"2020082514270474900_B54","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1109\/ICDM.2005.132","article-title":"Shortest-path kernels on graphs","volume-title":"Fifth IEEE International Conference on Data Mining","author":"Borgwardt,","year":"2005"},{"key":"2020082514270474900_B55","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/j.cad.2005.10.011","article-title":"Laplace-Beltrami spectra as \u2018Shape-DNA\u2019 of surfaces and solids","volume":"38","author":"Reuter,","year":"2006","journal-title":"Comput.-Aided Des."},{"key":"2020082514270474900_B56","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1109\/TCYB.2016.2526058","article-title":"Task sensitive feature exploration and learning for multitask graph classification","volume":"47","author":"Pan,","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"2020082514270474900_B57","doi-asserted-by":"crossref","first-page":"1914","DOI":"10.1002\/hbm.21333","article-title":"A whole brain fMRI atlas generated via spatially constrained spectral clustering","volume":"33","author":"Craddock,","year":"2012","journal-title":"Hum. Brain Map."}],"container-title":["Journal of Complex Networks"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/comnet\/article-pdf\/8\/3\/cnaa015\/33682081\/cnaa015.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/academic.oup.com\/comnet\/article-pdf\/8\/3\/cnaa015\/33682081\/cnaa015.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,25]],"date-time":"2020-08-25T18:27:23Z","timestamp":1598380043000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/comnet\/article\/doi\/10.1093\/comnet\/cnaa015\/5897078"}},"subtitle":[],"editor":[{"given":"Ernesto","family":"Estrada","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2020,6,1]]},"references-count":57,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,6,1]]}},"URL":"https:\/\/doi.org\/10.1093\/comnet\/cnaa015","relation":{},"ISSN":["2051-1310","2051-1329"],"issn-type":[{"type":"print","value":"2051-1310"},{"type":"electronic","value":"2051-1329"}],"subject":[],"published-other":{"date-parts":[[2020,6]]},"published":{"date-parts":[[2020,6,1]]},"article-number":"cnaa015"}}