{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:36:53Z","timestamp":1764571013290,"version":"3.37.3"},"reference-count":58,"publisher":"Oxford University Press (OUP)","issue":"8","license":[{"start":{"date-parts":[[2022,4,23]],"date-time":"2022-04-23T00:00:00Z","timestamp":1650672000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Shanghai Municipal Science and Technology Major Project","award":["2018SHZDZX01"],"award-info":[{"award-number":["2018SHZDZX01"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U20B2051","61872093"],"award-info":[{"award-number":["U20B2051","61872093"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fudan\u2019s Undergraduate Research Opportunities Program","award":["20001"],"award-info":[{"award-number":["20001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,8,14]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>It is well known that in many real networks, such as brain networks and scientific collaboration networks, there exist higher order nonpairwise relations among nodes, i.e. interactions between more than two nodes at a time. This simplicial structure can be described by simplicial complexes and has an important effect on topological and dynamical properties of networks involving such group interactions. In this paper, we study analytically resistance distances in iteratively growing networks with higher order interactions characterized by the simplicial structure that is controlled by a parameter $q$. We derive exact formulas for interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance, which have found applications in various areas elsewhere. We show that the average resistance distance tends to a $q$-dependent constant, indicating the impact of simplicial organization on the structural robustness measured by average resistance distance.<\/jats:p>","DOI":"10.1093\/comjnl\/bxac052","type":"journal-article","created":{"date-parts":[[2022,4,14]],"date-time":"2022-04-14T11:13:14Z","timestamp":1649934794000},"page":"1922-1935","source":"Crossref","is-referenced-by-count":4,"title":["Resistance Distances In Simplicial Networks"],"prefix":"10.1093","volume":"66","author":[{"given":"Mingzhe","family":"Zhu","sequence":"first","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing & Shanghai Engineering Research Institute of Blockchain , School of Computer Science, Fudan University, Shanghai 200433, China"}]},{"given":"Wanyue","family":"Xu","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing & Shanghai Engineering Research Institute of Blockchain , School of Computer Science, Fudan University, Shanghai 200433, China"}]},{"given":"Zhongzhi","family":"Zhang","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing & Shanghai Engineering Research Institute of Blockchain , School of Computer Science, Fudan University, Shanghai 200433, China"}]},{"given":"Haibin","family":"Kan","sequence":"additional","affiliation":[{"name":"Shanghai Key Laboratory of Intelligent Information Processing & Shanghai Engineering Research Institute of Blockchain , School of Computer Science, Fudan University, Shanghai 200433, China"}]},{"given":"Guanrong","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering , City University of Hong Kong, Hong Kong SAR, China"}]}],"member":"286","published-online":{"date-parts":[[2022,4,23]]},"reference":[{"key":"2023081805144402400_ref1","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."},{"volume-title":"Network Science","year":"2016","author":"Barab\u00e1si","key":"2023081805144402400_ref2"},{"key":"2023081805144402400_ref3","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."},{"key":"2023081805144402400_ref4","article-title":"Simplicial complexes and complex systems","volume":"40","author":"Salnikov","year":"2018","journal-title":"Eur. J. Phys."},{"key":"2023081805144402400_ref5","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1126\/science.aad9029","article-title":"Higher-order organization of complex networks","volume":"353","author":"Benson","year":"2016","journal-title":"Science"},{"key":"2023081805144402400_ref6","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1038\/nature23273","article-title":"Higher-order interactions stabilize dynamics in competitive network models","volume":"548","author":"Grilli","year":"2017","journal-title":"Nature"},{"key":"2023081805144402400_ref7","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":"2023081805144402400_ref8","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":"2023081805144402400_ref9","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."},{"key":"2023081805144402400_ref10","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":"2023081805144402400_ref11","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":"2023081805144402400_ref12","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.95.062301","article-title":"Weighted growing simplicial complexes","volume":"95","author":"Courtney","year":"2017","journal-title":"Phys. Rev. E"},{"key":"2023081805144402400_ref13","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.121.228301","article-title":"Simplicial activity driven model","volume":"121","author":"Petri","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"2023081805144402400_ref14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10827-016-0608-6","article-title":"Two\u2019s company, three (or more) is a simplex","volume":"41","author":"Giusti","year":"2016","journal-title":"J. Comput. Neurosci."},{"key":"2023081805144402400_ref15","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.97.052303","article-title":"Dense power-law networks and simplicial complexes","volume":"97","author":"Courtney","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2023081805144402400_ref16","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.97.032316","article-title":"Complex network view of evolving manifolds","volume":"97","author":"Silva","year":"2018","journal-title":"Phys. Rev. E"},{"key":"2023081805144402400_ref17","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":"2023081805144402400_ref18","doi-asserted-by":"crossref","first-page":"41974","DOI":"10.1038\/srep41974","article-title":"Emergent hyperbolic network geometry","volume":"7","author":"Bianconi","year":"2017","journal-title":"Sci. Rep."},{"key":"2023081805144402400_ref19","doi-asserted-by":"crossref","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":"2023081805144402400_ref20","doi-asserted-by":"crossref","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":"2023081805144402400_ref21","doi-asserted-by":"crossref","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. Research"},{"key":"2023081805144402400_ref22","article-title":"Modeling higher-order interactions in complex networks by edge product of graphs","author":"Wang","year":"2021","journal-title":"Comput. J."},{"key":"2023081805144402400_ref23","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1137\/080734029","article-title":"Graph sparsification by effective resistances","volume":"40","author":"Spielman","year":"2011","journal-title":"SIAM J. Comput."},{"key":"2023081805144402400_ref24","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1109\/TCSI.2012.2215780","article-title":"Kron reduction of graphs with applications to electrical networks","volume":"60","author":"Dorfler","year":"2013","journal-title":"IEEE Trans. Circuits Syst. I: Regular Papers"},{"key":"2023081805144402400_ref25","doi-asserted-by":"crossref","first-page":"1727","DOI":"10.1109\/TAC.2015.2481978","article-title":"A new notion of effective resistance for directed graphs\u2014Part I: Definition and properties","volume":"61","author":"Young","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"2023081805144402400_ref26","doi-asserted-by":"crossref","first-page":"1090","DOI":"10.1109\/TCSI.2018.2880601","article-title":"Network science meets circuit theory: Resistance distance, Kirchhoff index, and Foster\u2019s theorems with generalizations and unification. IEEE trans","volume":"66","author":"Thulasiraman","year":"2019","journal-title":"Circuits Syst. I: Regular Papers"},{"key":"2023081805144402400_ref27","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1109\/JPROC.2018.2821924","article-title":"Electrical networks and algebraic graph theory: Models, properties, and applications","volume":"106","author":"D\u00f6rfler","year":"2018","journal-title":"Proc. IEEE"},{"key":"2023081805144402400_ref28","doi-asserted-by":"crossref","first-page":"6898","DOI":"10.1109\/TIT.2019.2925610","article-title":"Low mean hitting time for random walks on heterogeneous networks","volume":"65","author":"Sheng","year":"2019","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2023081805144402400_ref29","doi-asserted-by":"crossref","first-page":"4415","DOI":"10.1109\/TCSI.2019.2929180","article-title":"On extension of effective resistance with application to graph laplacian definiteness and power network stability","volume":"66","author":"Song","year":"2019","journal-title":"IEEE Trans. Circuits Syst. I: Regular Papers"},{"key":"2023081805144402400_ref30","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.93.062311","article-title":"Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes","volume":"93","author":"Courtney","year":"2016","journal-title":"Phys. Rev. E"},{"key":"2023081805144402400_ref31","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.101.022308","article-title":"Random walks on hypergraphs","volume":"101","author":"Carletti","year":"2020","journal-title":"Phys. Rev. E"},{"key":"2023081805144402400_ref32","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/0024-3795(94)90084-1","article-title":"Reverse order laws for the generalized inverses of multiple matrix products","volume":"211","author":"Tian","year":"1994","journal-title":"Linear Algebra Appl."},{"key":"2023081805144402400_ref33","doi-asserted-by":"crossref","first-page":"36617","DOI":"10.1109\/ACCESS.2019.2902012","article-title":"The resistance distance and Kirchhoff index on quadrilateral graph and pentagonal graph","volume":"7","author":"Liu","year":"2019","journal-title":"IEEE Access"},{"key":"2023081805144402400_ref34","doi-asserted-by":"crossref","DOI":"10.5948\/UPO9781614440222","volume-title":"Random Walks and Electric Networks","author":"Doyle","year":"1984"},{"key":"2023081805144402400_ref35","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/BF01164627","article-title":"Resistance distance","volume":"12","author":"Klein","year":"1993","journal-title":"J. Math. Chem."},{"key":"2023081805144402400_ref36","first-page":"87","article-title":"Resistance distance in graphs","volume":"68","author":"Bapat","year":"1999","journal-title":"Math. Student"},{"key":"2023081805144402400_ref37","first-page":"633","article-title":"Resistance-distance sum rules","volume":"75","author":"Klein","year":"2002","journal-title":"Croat. Chem. Acta"},{"key":"2023081805144402400_ref38","first-page":"333","volume-title":"Contributions to Applied Mechanics (Reissner Anniversary Volume)?","author":"Foster","year":"1949"},{"key":"2023081805144402400_ref39","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1109\/TCT.1961.1086748","article-title":"An extension of a network theorem","volume":"8","author":"Foster","year":"1961","journal-title":"IRE Trans. Circuit Theory"},{"key":"2023081805144402400_ref40","doi-asserted-by":"crossref","first-page":"1691","DOI":"10.1016\/j.dam.2010.05.020","article-title":"Random walks and the effective resistance sum rules","volume":"158","author":"Chen","year":"2010","journal-title":"Discrete Appl. Math."},{"key":"2023081805144402400_ref41","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1137\/050645452","article-title":"Minimizing effective resistance of a graph","volume":"50","author":"Ghosh","year":"2008","journal-title":"SIAM Rev."},{"key":"2023081805144402400_ref42","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1109\/JSAC.2010.100105","article-title":"Autonomic traffic engineering for network robustness","volume":"28","author":"Tizghadam","year":"2010","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"2023081805144402400_ref43","first-page":"2377","volume-title":"Kirchhoff index as a measure of edge centrality in weighted networks: Nearly linear time algorithms","author":"Li","year":"2018"},{"key":"2023081805144402400_ref44","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1109\/TCNS.2014.2357552","article-title":"Consensus and coherence in fractal networks","volume":"1","author":"Patterson","year":"2014","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"2023081805144402400_ref45","doi-asserted-by":"crossref","first-page":"592","DOI":"10.1109\/TCYB.2017.2781714","article-title":"Consensus in self-similar hierarchical graphs and Sierpi\u0144ski graphs: Convergence speed, delay robustness, and coherence","volume":"49","author":"Qi","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"2023081805144402400_ref46","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":"2023081805144402400_ref47","doi-asserted-by":"crossref","first-page":"2024","DOI":"10.1109\/TCYB.2017.2726102","article-title":"Distributed parametric consensus optimization with an application to model predictive consensus problem","volume":"48","author":"Shi","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"2023081805144402400_ref48","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1109\/TCYB.2020.2965586","article-title":"Fast approximation of coherence for second-order noisy consensus networks","volume":"52","author":"Zhang","year":"2022","journal-title":"IEEE Trans. Cybern."},{"key":"2023081805144402400_ref49","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1109\/TIT.2021.3127272","article-title":"Biharmonic distance-based performance metric for second-order noisy consensus networks","volume":"68","author":"Yi","year":"2022","journal-title":"IEEE Trans. Inf. Theory"},{"key":"2023081805144402400_ref50","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":"2023081805144402400_ref51","first-page":"27","article-title":"Degree resistance distance of unicyclic graphs","volume":"1","author":"Gutman","year":"2012","journal-title":"Trans. Combin."},{"key":"2023081805144402400_ref52","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. \u2014 Theor. Methods"},{"key":"2023081805144402400_ref53","first-page":"46","volume-title":"Proceedings of The Web Conference","author":"Xu","year":"2020"},{"key":"2023081805144402400_ref54","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":"2023081805144402400_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":"2023081805144402400_ref56","doi-asserted-by":"crossref","first-page":"1389","DOI":"10.1109\/TAC.2018.2863203","article-title":"Scaling laws for consensus protocols subject to noise","volume":"64","author":"Jadbabaie","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"2023081805144402400_ref57","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1109\/TCST.2016.2550582","article-title":"Robustness of first-and second-order consensus algorithms for a noisy scale-free small-world Koch network","volume":"25","author":"Yi","year":"2017","journal-title":"IEEE Trans. Control Syst. Technol."},{"key":"2023081805144402400_ref58","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2020.05.004","article-title":"Networks beyond pairwise interactions: Structure and dynamics","volume":"874","author":"Battiston","year":"2020","journal-title":"Phys. Rep."}],"container-title":["The Computer Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/comjnl\/article-pdf\/66\/8\/1922\/51107480\/bxac052.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/comjnl\/article-pdf\/66\/8\/1922\/51107480\/bxac052.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,18]],"date-time":"2023-08-18T05:15:33Z","timestamp":1692335733000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/comjnl\/article\/66\/8\/1922\/6572654"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,23]]},"references-count":58,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,4,23]]},"published-print":{"date-parts":[[2023,8,14]]}},"URL":"https:\/\/doi.org\/10.1093\/comjnl\/bxac052","relation":{},"ISSN":["0010-4620","1460-2067"],"issn-type":[{"type":"print","value":"0010-4620"},{"type":"electronic","value":"1460-2067"}],"subject":[],"published-other":{"date-parts":[[2023,8]]},"published":{"date-parts":[[2022,4,23]]}}}