Structural and functional properties of spatially embedded scale-free networks.

Scale-free networks have been studied mostly as non-spatially embedded systems. However, in many realistic cases, they are spatially embedded and these constraints should be considered. Here, we study the structural and functional properties of a model of scale-free (SF) spatially embedded networks. In our model, both the degree and the length of links follow power law distributions as found in many real networks. We show that not all SF networks can be embedded in space and that the largest degree of a node in the network is usually smaller than in nonembedded SF networks. Moreover, the spatial constraints (each node has only few neighboring nodes) introduce degree-degree anticorrelations (disassortativity) since two high degree nodes cannot stay close in space. We also find significant effects of space embedding on the hopping distances (chemical distance) and the vulnerability of the networks.

[1]  J. Hopcroft,et al.  Are randomly grown graphs really random? , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[3]  J. S. Andrade,et al.  Optimal transport exponent in spatially embedded networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[5]  W. Marsden.  I and J , 2012 .

[6]  I M Sokolov,et al.  Evolving networks with disadvantaged long-range connections. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[8]  Daqing Li,et al.  Download details: IP Address: 129.74.250.206 , 2011 .

[9]  Cristian F. Moukarzel,et al.  Percolation in networks with long-range connections , 2006 .

[10]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[11]  Etienne Huens,et al.  Geographical dispersal of mobile communication networks , 2008, 0802.2178.

[12]  H E Stanley,et al.  Towards design principles for optimal transport networks. , 2010, Physical review letters.

[13]  Reuven Cohen,et al.  Scale-free networks on lattices. , 2002, Physical review letters.

[14]  P. Gács,et al.  Algorithms , 1992 .

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

[16]  Shlomo Havlin,et al.  Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance , 2012, ArXiv.

[17]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[18]  Albert-László Barabási,et al.  Internet: Diameter of the World-Wide Web , 1999, Nature.

[19]  Cristian F. Moukarzel,et al.  Effective dimensions in networks with long-range connections , 2005 .

[20]  S. Havlin,et al.  Structural properties of spatially embedded networks , 2008, 0804.2575.

[21]  Shlomo Havlin,et al.  Diffusion, annihilation, and chemical reactions in complex networks with spatial constraints. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[23]  Alessandro Vespignani,et al.  Dynamical Processes on Complex Networks , 2008 .

[24]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[25]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[26]  M. A. de Menezes,et al.  Shortest paths on systems with power-law distributed long-range connections. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  M E J Newman.  Assortative mixing in networks. , 2002, Physical review letters.

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

[29]  A. Barabasi,et al.  Scale-free characteristics of random networks: the topology of the world-wide web , 2000 .

[30]  Jon M. Kleinberg,et al.  Navigation in a small world , 2000, Nature.

[31]  L. Sander,et al.  Geography in a scale-free network model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  S. Havlin,et al.  Dimension of spatially embedded networks , 2011 .

[33]  P. Pin,et al.  Assessing the relevance of node features for network structure , 2008, Proceedings of the National Academy of Sciences.

[34]  Parongama Sen,et al.  Modulated scale-free network in Euclidean space. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Béla Bollobás,et al.  Random Graphs , 1985 .