Stefan Boettcher et al 2008 J. Phys. A: Math. Theor. 41 252001 doi:10.1088/1751-8113/41/25/252001
Hierarchical regular small-world networks
FREE ARTICLEStefan Boettcher1, Bruno Gonçalves1 and Hasan Guclu2
Show affiliationsFAST TRACK COMMUNICATION
Two new networks are introduced that resemble small-world properties. These networks are recursively constructed but retain a fixed, regular degree. They possess a unique one-dimensional lattice backbone overlaid by a hierarchical sequence of long-distance links, mixing real-space and small-world features. Both networks, one 3-regular and the other 4-regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3-regular network is planar, has a diameter growing as 
with system size N, and leads to super-diffusion with an exact, anomalous exponent dw = 1.306..., but possesses only a trivial fixed point Tc = 0 for the Ising ferromagnet. In turn, the 4-regular network is non-planar, has a diameter growing as 
, exhibits 'ballistic' diffusion (dw = 1), and a non-trivial ferromagnetic transition, Tc > 0. It suggests that the 3-regular network is still quite 'geometric', while the 4-regular network qualifies as a true small world with mean-field properties. As an engineering application we discuss synchronization of processors on these networks.
- PACS
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89.75.Hc Networks and genealogical trees
75.30.Kz Magnetic phase boundaries (including magnetic transitions, metamagnetism, etc.)
05.10.Cc Renormalization group methods
- MSC
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82C41 Dynamics of random walks, random surfaces, lattice animals, etc. (See also 60G50)
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs
82C26 Dynamic and nonequilibrium phase transitions (general)
82C28 Dynamic renormalization group methods (See also 81T17)
- Subjects
- Dates
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Issue 25 (27 June 2008)
Received 31 March 2008, in final form 1 April 2008
Published 27 May 2008
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Hierarchical regular small-world networks
Stefan Boettcher et al 2008 J. Phys. A: Math. Theor. 41 252001
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