A class of vertex–edge-growth small-world network models having scale-free, self-similar and hierarchical characters

Abstract The problem of uncovering the internal operating function of network models is intriguing, demanded and attractive in researches of complex networks. Notice that, in the past two decades, a great number of artificial models are built to try to answer the above mentioned task. Based on the different growth ways, these previous models can be divided into two categories, one type, possessing the preferential attachment, follows a power-law P ( k ) ∼ k − γ , 2 γ 3 . The other has exponential-scaling feature, P ( k ) ∼ α − k . However, there are no models containing above two kinds of growth ways to be presented, even the study of interconnection between these two growth manners in the same model is lacking. Hence, in this paper, we construct a class of planar and self-similar graphs motivated from a new attachment way, vertex–edge-growth network-operation, more precisely, the couple of both them. We report that this model is sparse, small world and hierarchical. And then, not only is scale-free feature in our model, but also lies the degree parameter γ ( ≈ 3 . 242 ) out the typical range. Note that, we suggest that the coexistence of multiple vertex growth ways will have a prominent effect on the power-law parameter γ , and the preferential attachment plays a dominate role on the development of networks over time. At the end of this paper, we obtain an exact analytical expression for the total number of spanning trees of models and also capture spanning trees entropy which we have compared with those of their corresponding component elements.

[1]  Fabricio A. Breve,et al.  Fuzzy community structure detection by particle competition and cooperation , 2012, Soft Computing.

[2]  Bin Wu,et al.  The number of spanning trees in Apollonian networks , 2012, Discret. Appl. Math..

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

[4]  Jing Huang,et al.  ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS , 2015, Bulletin of the Australian Mathematical Society.

[5]  Bin Wu,et al.  Counting spanning trees in a small-world Farey graph , 2012, 1201.4228.

[6]  Heiko Rieger,et al.  Random walks on complex networks. , 2004, Physical review letters.

[7]  R. Albert,et al.  The large-scale organization of metabolic networks , 2000, Nature.

[8]  A. Motter,et al.  Synchronization is optimal in nondiagonalizable networks. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Stephan G. Wagner,et al.  On the number of spanning trees on various lattices , 2010 .

[10]  Bin Wu,et al.  Counting spanning trees in self-similar networks by evaluating determinants , 2011, 1105.0565.

[11]  C. Lee Giles,et al.  Self-Organization and Identification of Web Communities , 2002, Computer.

[12]  Lev Guzmán-Vargas,et al.  Evaluating the transport in small-world and scale-free networks , 2014 .

[13]  Bin Liu,et al.  A scale-free topology model with fault-tolerance and intrusion-tolerance in wireless sensor networks , 2016, Comput. Electr. Eng..

[14]  F. Radicchi,et al.  Statistical significance of communities in networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M E Newman,et al.  Scientific collaboration networks. I. Network construction and fundamental results. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  S. Ostlund,et al.  Renormalisation-group calculations of finite systems: order parameter and specific heat for epitaxial ordering , 1979 .

[17]  Guanrong Chen,et al.  Random walks on weighted networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Reka Albert,et al.  Mean-field theory for scale-free random networks , 1999 .

[19]  R. Guimerà,et al.  Functional cartography of complex metabolic networks , 2005, Nature.

[20]  Ming-Yang Zhou,et al.  Traffic dynamics in scale-free networks with tunable strength of community structure , 2012 .

[21]  Chonghui Guo,et al.  A deterministic small-world network created by edge iterations , 2006 .

[22]  B. Kahng,et al.  Geometric fractal growth model for scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Bing Yao,et al.  Scale-free Network Models With Parameters , 2016 .

[24]  F. Y. Wu.  Number of spanning trees on a lattice , 1977 .

[25]  J. Slotine,et al.  Spectrum of controlling and observing complex networks , 2015, Nature Physics.

[26]  H. Ez-Zahraouy,et al.  Implantation of the global dynamic routing scheme in scale-free networks under the shortest path strategy , 2016 .

[27]  Philippe Marchal.  Loop-Erased Random Walks, Spanning Trees and Hamiltonian Cycles , 2000 .

[28]  Yuejin Tan,et al.  Heterogeneity of Scale-free Networks , 2007 .

[29]  Wei-Shih Yang,et al.  Spanning Trees on the Sierpinski Gasket , 2006, cond-mat/0609453.

[30]  S. N. Dorogovtsev,et al.  Pseudofractal scale-free web. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  A. Crisanti,et al.  Critical Study of Hierarchical Lattice Renormalization Group in Magnetic Ordered and Quenched Disordered Systems: Ising and Blume–Emery–Griffiths Models , 2014 .

[32]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[33]  Bing Yao,et al.  Non-planar unclustered Peterson graphs as scale-free models of the Internet of Things , 2016, 2016 IEEE Information Technology, Networking, Electronic and Automation Control Conference.

[34]  Luciano da Fontoura Costa,et al.  Probing the Statistical Properties of Unknown Texts: Application to the Voynich Manuscript , 2013, PloS one.

[35]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[36]  Zhongzhi Zhang,et al.  The Number of Spanning Trees of an Infinite Family of Outerplanar, Small-World and Self-Similar Graphs , 2012, ArXiv.

[37]  Yang Wang,et al.  The effect of hub nodes on the community structure in scale-free networks , 2011 .

[38]  M. Kaufman,et al.  Spin systems on hierarchical lattices. Introduction and thermodynamic limit , 1982 .

[39]  K-I Goh,et al.  Skeleton and fractal scaling in complex networks. , 2006, Physical review letters.

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

[41]  F. Chung,et al.  The average distances in random graphs with given expected degrees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[42]  Takashi Tahara,et al.  The networks from medical knowledge and clinical practice have small-world, scale-free, and hierarchical features , 2013 .

[43]  E. Agliari,et al.  Random walks on deterministic scale-free networks: exact results. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Albert-Laszlo Barabasi,et al.  Deterministic scale-free networks , 2001 .

[45]  Francesc Comellas,et al.  Label-based routing for a family of scale-free, modular, planar and unclustered graphs , 2011 .

[46]  Petter Holme,et al.  Subnetwork hierarchies of biochemical pathways , 2002, Bioinform..

[47]  Shuigeng Zhou,et al.  Enumeration of spanning trees in a pseudofractal scale-free web , 2010, 1008.0267.

[48]  Bing Yao,et al.  AS-mixed Network Models Created By Triangle-expanding Operations , 2016 .

[49]  János Kertész,et al.  Geometry of minimum spanning trees on scale-free networks , 2003 .

[50]  Dinghua Shi,et al.  The modeling of scale-free networks☆ , 2004 .

[51]  F. Y. Wu,et al.  Spanning trees on graphs and lattices in d dimensions , 2000, cond-mat/0004341.

[52]  S. Havlin,et al.  Scale-free networks are ultrasmall. , 2002, Physical review letters.

[53]  A. Wagner.  The yeast protein interaction network evolves rapidly and contains few redundant duplicate genes. , 2001, Molecular biology and evolution.