{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T20:12:37Z","timestamp":1780431157343,"version":"3.54.1"},"reference-count":72,"publisher":"Oxford University Press (OUP)","issue":"9","license":[{"start":{"date-parts":[[2021,6,22]],"date-time":"2021-06-22T00:00:00Z","timestamp":1624320000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61803248"],"award-info":[{"award-number":["61803248"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61872093"],"award-info":[{"award-number":["61872093"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U19A2066"],"award-info":[{"award-number":["U19A2066"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U20B2051"],"award-info":[{"award-number":["U20B2051"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2018YFB1305104"],"award-info":[{"award-number":["2018YFB1305104"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2019YFB2101703"],"award-info":[{"award-number":["2019YFB2101703"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shanghai Municipal Science and Technology Major Project","award":["2018SHZDZX01"],"award-info":[{"award-number":["2018SHZDZX01"]}]},{"name":"Shanghai Municipal Science and Technology Major Project","award":["2021SHZDZX03"],"award-info":[{"award-number":["2021SHZDZX03"]}]},{"name":"Fudan Undergraduate Research Opportunities Program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,9,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Many graph products have been applied to generate complex networks with striking properties observed in real-world systems. In this paper, we propose a simple generative model for simplicial networks by iteratively using edge corona product. We present a comprehensive analysis of the structural properties of the network model, including degree distribution, diameter, clustering coefficient, as well as distribution of clique sizes, obtaining explicit expressions for these relevant quantities, which agree with the behaviors found in diverse real networks. Moreover, we obtain exact expressions for all the eigenvalues and their associated multiplicities of the normalized Laplacian matrix, based on which we derive explicit formulas for mixing time, mean hitting time and the number of spanning trees. Thus, as previous models generated by other graph products, our model is also an exactly solvable one, whose structural properties can be analytically treated. More interestingly, the expressions for the spectra of our model are also exactly determined, which is sharp contrast to previous models whose spectra can only be given recursively at most. This advantage makes our model a good test bed and an ideal substrate network for studying dynamical processes, especially those closely related to the spectra of normalized Laplacian matrix, in order to uncover the influences of simplicial structure on these processes.<\/jats:p>","DOI":"10.1093\/comjnl\/bxab070","type":"journal-article","created":{"date-parts":[[2021,5,6]],"date-time":"2021-05-06T11:34:08Z","timestamp":1620300848000},"page":"2347-2359","source":"Crossref","is-referenced-by-count":6,"title":["Modeling Higher-Order Interactions in Complex Networks by Edge Product of Graphs"],"prefix":"10.1093","volume":"65","author":[{"given":"Yucheng","family":"Wang","sequence":"first","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing , School of Computer Science, Fudan University, Shanghai 200433, China; Shanghai Engineering Research Institute of Blockchain, Shanghai 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuhao","family":"Yi","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing , School of Computer Science, Fudan University, Shanghai 200433, China; Shanghai Engineering Research Institute of Blockchain, Shanghai 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wanyue","family":"Xu","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing , School of Computer Science, Fudan University, Shanghai 200433, China; Shanghai Engineering Research Institute of Blockchain, Shanghai 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhongzhi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing , School of Computer Science, Fudan University, Shanghai 200433, China; Shanghai Engineering Research Institute of Blockchain, Shanghai 200433, China; Research Institute of Intelligent Complex Systems, Fudan University, Shanghai 200433, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2021,6,22]]},"reference":[{"key":"2022091610474598400_ref1","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/BF01048272","article-title":"Reversible Markov chains and random walks on graphs","volume":"2","author":"Aldous","year":"1993","journal-title":"J. Theor. Probab."},{"key":"2022091610474598400_ref2","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":"2022091610474598400_ref3","doi-asserted-by":"crossref","first-page":"3747","DOI":"10.1073\/pnas.0400087101","article-title":"The architecture of complex weighted networks","volume":"101","author":"Barrat","year":"2004","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2022091610474598400_ref4","doi-asserted-by":"crossref","first-page":"225202","DOI":"10.1088\/1751-8113\/49\/22\/225202","article-title":"Deterministic hierarchical networks","volume":"49","author":"Barriere","year":"2016","journal-title":"J. Phys. A"},{"key":"2022091610474598400_ref5","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.dam.2008.04.018","article-title":"The hierarchical product of graphs","volume":"157","author":"Barriere","year":"2009","journal-title":"Discrete Appl. Math"},{"key":"2022091610474598400_ref6","doi-asserted-by":"crossref","first-page":"3871","DOI":"10.1016\/j.disc.2008.10.028","article-title":"The generalized hierarchical product of graphs","volume":"309","author":"Barri\u00e8re","year":"2009","journal-title":"Discrete Math."},{"key":"2022091610474598400_ref7","doi-asserted-by":"crossref","first-page":"E11221","DOI":"10.1073\/pnas.1800683115","article-title":"Simplicial closure and higher-order link prediction","volume":"115","author":"Benson","year":"2018","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2022091610474598400_ref8","doi-asserted-by":"crossref","first-page":"062302","DOI":"10.1103\/PhysRevE.94.062302","article-title":"Densification and structural transitions in networks that grow by node copying","volume":"94","author":"Bhat","year":"2016","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref9","doi-asserted-by":"crossref","first-page":"014005","DOI":"10.1088\/1742-5468\/ab5d0e","article-title":"The spectral dimension of simplicial complexes: a renormalization group theory","volume":"2020","author":"Bianconi","year":"2020","journal-title":"J. Stat. Mech"},{"key":"2022091610474598400_ref10","doi-asserted-by":"crossref","first-page":"052308","DOI":"10.1103\/PhysRevE.98.052308","article-title":"Topological percolation on hyperbolic simplicial complexes","volume":"98","author":"Bianconi","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref11","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1145\/1284680.1284681","article-title":"Epidemic thresholds in real networks","volume":"10","author":"Chakrabarti","year":"2008","journal-title":"ACM Trans. Inform. Syst. Secur"},{"key":"2022091610474598400_ref12","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1016\/j.dam.2006.09.008","article-title":"Resistance distance and the normalized Laplacian spectrum","volume":"155","author":"Chen","year":"2007","journal-title":"Discrete. Appl. Math"},{"key":"2022091610474598400_ref13","doi-asserted-by":"crossref","first-page":"15879","DOI":"10.1073\/pnas.252631999","article-title":"The average distances in random graphs with given expected degrees","volume":"99","author":"Chung","year":"2002","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2022091610474598400_ref14","first-page":"92","article-title":"Spectral graph theory, regional conference series in math","author":"Chung","year":"1997","journal-title":"CBMS Amer. Math. Soc"},{"key":"2022091610474598400_ref15","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1038\/nature06201","article-title":"First-passage times in complex scale-invariant media","volume":"450","author":"Condamin","year":"2007","journal-title":"Nature"},{"key":"2022091610474598400_ref16","doi-asserted-by":"crossref","first-page":"062301","DOI":"10.1103\/PhysRevE.95.062301","article-title":"Weighted growing simplicial complexes","volume":"95","author":"Courtney","year":"2017","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref17","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1093\/comnet\/cny036","article-title":"The simplex geometry of graphs","volume":"7","author":"Devriendt","year":"2019","journal-title":"J. Complex Netw."},{"key":"2022091610474598400_ref18","doi-asserted-by":"crossref","first-page":"066122","DOI":"10.1103\/PhysRevE.65.066122","article-title":"Pseudofractal scale-free web","volume":"65","author":"Dorogovtsev","year":"2002","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref19","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1007\/BF01844162","article-title":"On the corona of two graphs","volume":"4","author":"Frucht","year":"1970","journal-title":"Aequationes Math."},{"key":"2022091610474598400_ref20","doi-asserted-by":"crossref","first-page":"066118","DOI":"10.1103\/PhysRevE.79.066118","article-title":"Random hypergraphs and their applications","volume":"79","author":"Ghoshal","year":"2009","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref21","doi-asserted-by":"crossref","first-page":"7821","DOI":"10.1073\/pnas.122653799","article-title":"Community structure in social and biological networks","volume":"99","author":"Girvan","year":"2002","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2022091610474598400_ref22","doi-asserted-by":"crossref","first-page":"13455","DOI":"10.1073\/pnas.1506407112","article-title":"Clique topology reveals intrinsic geometric structure in neural correlations","volume":"112","author":"Giusti","year":"2015","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2022091610474598400_ref23","volume-title":"Algebraic Topology","author":"Hatcher","year":"2002"},{"key":"2022091610474598400_ref24","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1002\/net.3230230504","article-title":"Applications of E-graphs in network design","volume":"23","author":"Haynes","year":"1993","journal-title":"Networks"},{"key":"2022091610474598400_ref25","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1080\/00207169508804359","article-title":"Invariants of E-graphs","volume":"55","author":"Haynes","year":"1995","journal-title":"Int. J. Comput. Math."},{"key":"2022091610474598400_ref26","first-page":"1","article-title":"Minimum Time Length Scheduling Under Blockage and Interference in Multi-hop Mmwave Networks","volume-title":"Proc. 2015 IEEE Global Communications Conf.","author":"He","year":"2015"},{"key":"2022091610474598400_ref27","doi-asserted-by":"crossref","first-page":"022305","DOI":"10.1103\/PhysRevE.101.022305","article-title":"Adaptive voter model on simplicial complexes","volume":"101","author":"Horstmeyer","year":"2020","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref28","doi-asserted-by":"crossref","first-page":"586","DOI":"10.13001\/1081-3810.1395","article-title":"The spectrum of the edge corona of two graphs","volume":"20","author":"Hou","year":"2010","journal-title":"Electron. J. Linear Algebra"},{"key":"2022091610474598400_ref29","doi-asserted-by":"crossref","first-page":"1309","DOI":"10.1080\/03610926.2012.741742","article-title":"The role of Kemeny\u2019s constant in properties of Markov chains","volume":"43","author":"Hunter","year":"2014","journal-title":"Commun. Stat. Theor. Methods"},{"key":"2022091610474598400_ref30","volume-title":"Product Graphs: Structure and Recognition","author":"Imrich","year":"2000"},{"key":"2022091610474598400_ref31","first-page":"441","article-title":"A Fast and Provable Method for Estimating Clique Counts using Tur\u00e1n\u2019s Theorem","volume-title":"Proc. 26th International Conf. World Wide Web","author":"Jain","year":"2017"},{"key":"2022091610474598400_ref32","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.tcs.2017.08.024","article-title":"Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs","volume":"703","author":"Jin","year":"2017","journal-title":"Theoret. Comput. Sci"},{"key":"2022091610474598400_ref33","volume-title":"Finite Markov Chains","author":"Kemeny","year":"1976"},{"key":"2022091610474598400_ref34","doi-asserted-by":"crossref","DOI":"10.1371\/journal.pcbi.1000385","article-title":"Hypergraphs and cellular networks","volume":"5","author":"Klamt","year":"2009","journal-title":"PLoS Comput. Biol"},{"key":"2022091610474598400_ref35","doi-asserted-by":"crossref","first-page":"025101","DOI":"10.1103\/PhysRevE.73.025101","article-title":"Statistics of cycles in large networks","volume":"73","author":"Klemm","year":"2006","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref36","doi-asserted-by":"crossref","first-page":"2485","DOI":"10.1038\/s41467-019-10431-6","article-title":"Simplicial models of social contagion","volume":"10","author":"Lacopini","year":"2019","journal-title":"Nat. Commun."},{"key":"2022091610474598400_ref37","first-page":"985","article-title":"Kronecker graphs: an approach to modeling networks","volume":"11","author":"Leskovec","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"2022091610474598400_ref38","first-page":"497","article-title":"Scalable Modeling of Real Graphs Using Kronecker Multiplication","volume-title":"Proc. 24th Int. Conf. Machine Learning","author":"Leskovec","year":"2007"},{"key":"2022091610474598400_ref39","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1080\/00029890.2002.11919905","article-title":"Kemeny\u2019s constant and the random surfer","volume":"109","author":"Levene","year":"2002","journal-title":"Am. Math. Mon"},{"key":"2022091610474598400_ref40","doi-asserted-by":"crossref","DOI":"10.1090\/mbk\/058","volume-title":"Markov Chains and Mixing Times","author":"Levin","year":"2008"},{"key":"2022091610474598400_ref41","first-page":"1","article-title":"Random walks on graphs: a survey","volume":"8","author":"Lov\u00e0sz","year":"1996","journal-title":"Combinatorics"},{"key":"2022091610474598400_ref42","doi-asserted-by":"crossref","first-page":"P11024","DOI":"10.1088\/1742-5468\/2015\/11\/P11024","article-title":"Corona graphs as a model of small-world networks","volume":"2015","author":"Lv","year":"2015","journal-title":"J. Stat. Mech. Theory. E"},{"key":"2022091610474598400_ref43","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/978-3-540-77004-6_14","article-title":"Stochastic Kronecker Graphs","volume-title":"Proc. 5th Int. Conf. Algorithms and Models for the Web-Graph","author":"Mahdian","year":"2007"},{"key":"2022091610474598400_ref44","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1002\/rsa.20335","article-title":"Stochastic Kronecker graphs","volume":"38","author":"Mahdian","year":"2011","journal-title":"Random Struct. Algorithms"},{"key":"2022091610474598400_ref45","doi-asserted-by":"crossref","first-page":"012049","DOI":"10.1103\/PhysRevResearch.2.012049","article-title":"Abrupt phase transition of epidemic spreading in simplicial complexes","volume":"2","author":"Matamalas","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"2022091610474598400_ref46","doi-asserted-by":"crossref","first-page":"022307","DOI":"10.1103\/PhysRevE.99.022307","article-title":"Synchronization in network geometries with finite spectral dimension","volume":"99","author":"Mill\u00e1n","year":"2019","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref47","doi-asserted-by":"crossref","first-page":"824","DOI":"10.1126\/science.298.5594.824","article-title":"Network motifs: simple building blocks of complex networks","volume":"298","author":"Milo","year":"2002","journal-title":"Science"},{"key":"2022091610474598400_ref48","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1145\/2783258.2783385","article-title":"Scalable Large Near-Clique Detection in Large-Scale Networks via Sampling","volume-title":"Proc. 21th ACM SIGKDD Int. Conf. Knowledge Discovery and Data Mining","author":"Mitzenmacher","year":"2015"},{"key":"2022091610474598400_ref49","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1145\/1879141.1879191","article-title":"Measuring the Mixing Time of Social Graphs","volume-title":"Proc. 10th ACM SIGCOMM Conf. Internet Measurement","author":"Mohaisen","year":"2010"},{"key":"2022091610474598400_ref50","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","author":"Newman","year":"2003","journal-title":"SIAM Rev."},{"key":"2022091610474598400_ref51","doi-asserted-by":"crossref","first-page":"118701","DOI":"10.1103\/PhysRevLett.92.118701","article-title":"Random walks on complex networks","volume":"92","author":"Noh","year":"2004","journal-title":"Phys. Rev. Lett"},{"key":"2022091610474598400_ref52","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1016\/j.jfa.2008.11.001","article-title":"Ricci curvature of markov chains on metric spaces","volume":"256","author":"Ollivier","year":"2009","journal-title":"J. Funct. Anal."},{"key":"2022091610474598400_ref53","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1109\/ICNP.2011.6089084","article-title":"Generalized Graph Products for Network Design and Analysis","volume-title":"2011 19th IEEE Int. Conf. Network Protocols","author":"Parsonage","year":"2011"},{"key":"2022091610474598400_ref54","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1140\/epjds\/s13688-017-0114-8","article-title":"The shape of collaborations","volume":"6","author":"Patania","year":"2017","journal-title":"EPJ Data Sci."},{"key":"2022091610474598400_ref55","doi-asserted-by":"crossref","first-page":"3156","DOI":"10.1109\/TAC.2015.2426317","article-title":"Robotic surveillance and Markov chains with minimal weighted Kemeny constant","volume":"60","author":"Patel","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"2022091610474598400_ref56","doi-asserted-by":"crossref","first-page":"228301","DOI":"10.1103\/PhysRevLett.121.228301","article-title":"Simplicial activity driven model","volume":"121","author":"Petri","year":"2018","journal-title":"Phys. Rev. Lett"},{"key":"2022091610474598400_ref57","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1093\/comjnl\/bxx094","article-title":"Extended corona product as an exactly tractable model for weighted heterogeneous networks","volume":"61","author":"Qi","year":"2018","journal-title":"Comput. J."},{"key":"2022091610474598400_ref58","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1093\/comjnl\/bxy087","article-title":"Topological and spectral properties of small-world hierarchical graphs","volume":"62","author":"Qi","year":"2019","journal-title":"Comput. J."},{"key":"2022091610474598400_ref59","doi-asserted-by":"crossref","first-page":"026112","DOI":"10.1103\/PhysRevE.67.026112","article-title":"Hierarchical organization in complex networks","volume":"67","author":"Ravasz","year":"2003","journal-title":"Phys. Rev. E"},{"key":"2022091610474598400_ref60","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511606014","volume-title":"A Guide to First-Passage Processes","author":"Redner","year":"2001"},{"key":"2022091610474598400_ref61","doi-asserted-by":"crossref","first-page":"48","DOI":"10.3389\/fncom.2017.00048","article-title":"Cliques of neurons bound into cavities provide a missing link between structure and function","volume":"11","author":"Reimann","year":"2017","journal-title":"Front. Comput. Neurosci"},{"key":"2022091610474598400_ref62","doi-asserted-by":"crossref","first-page":"4589","DOI":"10.1088\/0305-4470\/38\/21\/005","article-title":"Statistics of cycles: how loopy is your network?","volume":"38","author":"Rozenfeld","year":"2005","journal-title":"J. Phys. A"},{"key":"2022091610474598400_ref63","doi-asserted-by":"crossref","first-page":"014001","DOI":"10.1088\/1361-6404\/aae790","article-title":"Simplicial complexes and complex systems","volume":"40","author":"Salnikov","year":"2018","journal-title":"Eur. J. Phys."},{"key":"2022091610474598400_ref64","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1017\/S0963548300000390","article-title":"Improved bounds for mixing rates of Markov chains and multicommodity flow","volume":"1","author":"Sinclair","year":"1992","journal-title":"Combin. Probab. Comput"},{"key":"2022091610474598400_ref65","doi-asserted-by":"crossref","first-page":"248301","DOI":"10.1103\/PhysRevLett.122.248301","article-title":"Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes","volume":"122","author":"Skardal","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"2022091610474598400_ref66","first-page":"1122","article-title":"The K Clique Densest Subgraph Problem","volume-title":"Proc. 24th Int. Conf. World Wide Web","author":"Tsourakakis","year":"2015"},{"key":"2022091610474598400_ref67","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":"2022091610474598400_ref68","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1090\/S0002-9939-1962-0133816-6","article-title":"The Kronecker product of graphs","volume":"13","author":"Weichsel","year":"1962","journal-title":"Proc. Am. Math. Soc."},{"key":"2022091610474598400_ref69","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1038\/ng1242","article-title":"Evolutionary conservation of motif constituents in the yeast protein interaction network","volume":"35","author":"Wuchty","year":"2003","journal-title":"Nat. Genet."},{"key":"2022091610474598400_ref70","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1109\/TCYB.2018.2868124","article-title":"Scale-free loopy structure is resistant to noise in consensus dynamics in complex networks","volume":"50","author":"Yi","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"2022091610474598400_ref71","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1016\/j.physa.2006.11.006","article-title":"A general geometric growth model for pseudofractal scale-free web","volume":"377","author":"Zhang","year":"2007","journal-title":"Phys. A"},{"key":"2022091610474598400_ref72","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.tcs.2014.12.024","article-title":"Pfaffian orientations and perfect matchings of scale-free networks","volume":"570","author":"Zhang","year":"2015","journal-title":"Theoret. Comput. Sci"}],"container-title":["The Computer Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/comjnl\/article-pdf\/65\/9\/2347\/45882135\/bxab070.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/comjnl\/article-pdf\/65\/9\/2347\/45882135\/bxab070.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T10:49:50Z","timestamp":1663325390000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/comjnl\/article\/65\/9\/2347\/6307490"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,22]]},"references-count":72,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,6,22]]},"published-print":{"date-parts":[[2022,9,16]]}},"URL":"https:\/\/doi.org\/10.1093\/comjnl\/bxab070","relation":{},"ISSN":["0010-4620","1460-2067"],"issn-type":[{"value":"0010-4620","type":"print"},{"value":"1460-2067","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,9]]},"published":{"date-parts":[[2021,6,22]]}}}