The Kronecker-clique model for higher-order clustering coefficients

Abstract We propose a Kronecker-clique model, which possesses the higher-order properties, i.e., high-order clustering coefficients, of real-world networks. The higher-order clustering coefficient is defined as the closure probability of cliques. The higher-order structure of Kronecker-clique model is formed by introducing some cliques into the stochastic Kronecker model according to the degree-dependent function. We compare the higher-order clustering coefficients of the Kronecker-clique model with those of the stochastic Kronecker model and the HyperKron model when fitting the real-world networks. The results indicate that the Kronecker-clique model performs better than the stochastic Kronecker model, the HyperKron model as well as the traditional clustered model. Moreover, we perform k -core decomposition and show that the maximum k -core of the Kronecker-clique model is closer to that of real-world networks compared with the stochastic Kronecker model.

[1]  S. Shen-Orr,et al.  Superfamilies of Evolved and Designed Networks , 2004, Science.

[2]  S. Melnik,et al.  Analytical results for bond percolation and k-core sizes on clustered networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Guy Bresler,et al.  Mixing Time of Exponential Random Graphs , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[4]  P. Schwille,et al.  Discovery of 505-million-year old chitin in the basal demosponge Vauxia gracilenta , 2013, Scientific Reports.

[5]  Tamara G. Kolda,et al.  A Scalable Generative Graph Model with Community Structure , 2013, SIAM J. Sci. Comput..

[6]  A Díaz-Guilera,et al.  Self-similar community structure in a network of human interactions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Tamara G. Kolda,et al.  An in-depth analysis of stochastic Kronecker graphs , 2011, JACM.

[8]  Mason A. Porter,et al.  Social Structure of Facebook Networks , 2011, ArXiv.

[9]  Dag L. Aksnes,et al.  Mesopelagic fish biomass and trophic efficiency of the open ocean , 2014 .

[10]  Suresh Jagannathan,et al.  CompCertTSO: A Verified Compiler for Relaxed-Memory Concurrency , 2013, JACM.

[11]  Jean-Gabriel Young,et al.  Networks beyond pairwise interactions: structure and dynamics , 2020, ArXiv.

[12]  Linyuan Lu,et al.  High-Ordered Random Walks and Generalized Laplacians on Hypergraphs , 2011, WAW.

[13]  Cliff Joslyn,et al.  Hypernetwork science via high-order hypergraph walks , 2019, EPJ Data Science.

[14]  Rik Sarkar,et al.  Characteristic Functions on Graphs: Birds of a Feather, from Statistical Descriptors to Parametric Models , 2020, CIKM.

[15]  Ruyin Chen,et al.  Effect of external periodic regulations on Brownian motor , 2015 .

[16]  Pol Colomer-de-Simon,et al.  Deciphering the global organization of clustering in real complex networks , 2013, Scientific Reports.

[17]  Martin Rosvall,et al.  Memory in network flows and its effects on spreading dynamics and community detection , 2013, Nature Communications.

[18]  Harry Eugene Stanley,et al.  Correlation between centrality metrics and their application to the opinion model , 2014, The European Physical Journal B.

[19]  Yuval Shavitt,et al.  A model of Internet topology using k-shell decomposition , 2007, Proceedings of the National Academy of Sciences.

[20]  Zoran Levnajic,et al.  Revealing the Hidden Language of Complex Networks , 2014, Scientific Reports.

[21]  Jure Leskovec,et al.  Higher-order organization of complex networks , 2016, Science.

[22]  Yoram Louzoun,et al.  Mid size cliques are more common in real world networks than triangles , 2014, Network Science.

[23]  David F. Gleich,et al.  The HyperKron Graph Model for Higher-Order Features , 2018, 2018 IEEE International Conference on Data Mining (ICDM).

[24]  Christos Faloutsos,et al.  Scalable modeling of real graphs using Kronecker multiplication , 2007, ICML '07.

[25]  Christos Faloutsos,et al.  Kronecker Graphs: An Approach to Modeling Networks , 2008, J. Mach. Learn. Res..

[26]  Jennifer Neville,et al.  Tied Kronecker product graph models to capture variance in network populations , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[27]  Kathryn B. Laskey,et al.  Stochastic blockmodels: First steps , 1983 .

[28]  David F. Gleich,et al.  Moment-Based Estimation of Stochastic Kronecker Graph Parameters , 2011, Internet Math..

[29]  Fan Chung Graham,et al.  A random graph model for massive graphs , 2000, STOC '00.

[30]  Jure Leskovec,et al.  Higher-order clustering in networks , 2017, Physical review. E.

[31]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[32]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[33]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[34]  T. Kenny,et al.  CORRIGENDUM: Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators , 2014, Scientific Reports.

[35]  Sergey N. Dorogovtsev,et al.  K-core Organization of Complex Networks , 2005, Physical review letters.

[36]  Xiang Li,et al.  Mining the rank of universities with Wikipedia , 2018, Science China Information Sciences.