Emergence of long-range correlations in random networks

We perform an analytical analysis of the long-range degree correlation of the giant component in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the giant component is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the giant component becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of the $l$-distant node pairs, which behaves as an exponentially decreasing function of distance in the off-critical region. The correlation function obeys a power-law with an exponential cutoff near the critical point. The Erdős-Renyi random graph is employed to confirm this critical behavior.

[1]  T. E. Harris,et al.  The Theory of Branching Processes. , 1963 .

[2]  S. Havlin,et al.  Fractals and Disordered Systems , 1991 .

[3]  B. M. Fulk.  MATH , 1992 .

[4]  J. Herskowitz,et al.  Proceedings of the National Academy of Sciences, USA , 1996, Current Biology.

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

[6]  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.

[7]  Albert-László Barabási,et al.  Hierarchical organization in complex networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[9]  Claudio Castellano,et al.  Defining and identifying communities in networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[10]  A Vázquez,et al.  The topological relationship between the large-scale attributes and local interaction patterns of complex networks , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[11]  I. M. Sokolov,et al.  Construction and properties of assortative random networks , 2004 .

[12]  Marián Boguñá,et al.  Tuning clustering in random networks with arbitrary degree distributions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  S. Havlin,et al.  Self-similarity of complex networks , 2005, Nature.

[14]  A. Vázquez,et al.  Network clustering coefficient without degree-correlation biases. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[16]  Hildegard Meyer-Ortmanns,et al.  Self-similar scale-free networks and disassortativity. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[18]  Shlomo Havlin,et al.  Origins of fractality in the growth of complex networks , 2005, cond-mat/0507216.

[19]  E. Todeva.  Networks , 2007 .

[20]  S. Havlin,et al.  Scaling theory of transport in complex biological networks , 2007, Proceedings of the National Academy of Sciences.

[21]  Guido Caldarelli,et al.  Scale-Free Networks , 2007 .

[22]  Guido Caldarelli,et al.  Scale-Free Networks , 2007 .

[23]  Andrzej K Oleś,et al.  Correlations in connected random graphs. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Hernán A Makse,et al.  Scaling of degree correlations and its influence on diffusion in scale-free networks. , 2008, Physical review letters.

[25]  A. Arenas,et al.  Motif-based communities in complex networks , 2007, 0710.0059.

[26]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[27]  F Kawasaki,et al.  Reciprocal relation between the fractal and the small-world properties of complex networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Hernán D. Rozenfeld,et al.  Quantifying long-range correlations in complex networks beyond nearest neighbors , 2010, 1005.3680.

[29]  Andrzej K Oleś,et al.  Long-range disassortative correlations in generic random trees. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Reuven Cohen,et al.  Complex Networks: Structure, Robustness and Function , 2010 .

[31]  Hernán A Makse,et al.  Small-world to fractal transition in complex networks: a renormalization group approach. , 2009, Physical review letters.

[32]  Mason A. Porter,et al.  Core-Periphery Structure in Networks , 2012, SIAM J. Appl. Math..

[33]  Chiara Orsini,et al.  How random are complex networks , 2015, ArXiv.

[34]  Ivan Kryven,et al.  General expression for the component size distribution in infinite configuration networks. , 2017, Physical review. E.

[35]  Yuka Fujiki,et al.  Fractality and degree correlations in scale-free networks , 2017, The European Physical Journal B.

[36]  A J E M Janssen,et al.  Clustering spectrum of scale-free networks. , 2017, Physical review. E.

[37]  Takehisa Hasegawa,et al.  Disassortativity of percolating clusters in random networks , 2018, Physical Review E.

[38]  Reimer Kühn,et al.  Revealing the microstructure of the giant component in random graph ensembles. , 2018, Physical review. E.

[39]  Taro Takaguchi,et al.  General formulation of long-range degree correlations in complex networks. , 2017, Physical review. E.

[40]  Antonio-José Almeida,et al.  NAT , 2019, Springer Reference Medizin.

[41]  Yuka Fujiki,et al.  Identification of intrinsic long-range degree correlations in complex networks , 2019, Physical review. E.

[42]  Takehisa Hasegawa,et al.  Structure of percolating clusters in random clustered networks. , 2019, Physical review. E.

[43]  P. Alam.  ‘G’ , 2021, Composites Engineering: An A–Z Guide.

[44]  P. Alam,et al.  R , 1823, The Herodotus Encyclopedia.