Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2007-12-08
J. Phys. A: Math. Theor. 41, 252001 (2008)
Physics
Condensed Matter
Disordered Systems and Neural Networks
9 pages, final version for JPA FastTrack, find related articles at http://www.physics.emory.edu/faculty/boettcher
Scientific paper
10.1088/1751-8113/41/25/252001
Two new classes of networks are introduced that resemble small-world properties. These networks are recursively constructed but retain a fixed, regular degree. They consist of a one-dimensional lattice backbone overlayed by a hierarchical sequence of long-distance links. Both types of networks, one 3-regular and the other 4-regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3-regular networks are planar, have a diameter growing as \sqrt{N} with the system size N, and lead to super-diffusion with an exact, anomalous exponent d_w=1.3057581..., but possesses only a trivial fixed point T_c=0 for the Ising ferromagnet. In turn, the 4-regular networks are non-planar, have a diameter growing as ~2^[\sqrt(\log_2 N^2)], exhibit "ballistic" diffusion (d_w=1), and a non-trivial ferromagnetic transition, T_c>0. It suggest that the 3-regular networks are still quite "geometric", while the 4-regular networks qualify as true small-world networks with mean-field properties. As an example of an application we discuss synchronization of processors on these networks.
Boettcher Stefan
Goncalves Bruno
Guclu Hasan
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