Hierarchical, Regular SmallWorld Networks
Abstract
Two new classes of networks are introduced that resemble smallworld properties. These networks are recursively constructed but retain a fixed, regular degree. They consist of a onedimensional lattice backbone overlayed by a hierarchical sequence of longdistance links. Both types of networks, one 3regular and the other 4regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3regular networks are planar, have a diameter growing as \sqrt{N} with the system size N, and lead to superdiffusion 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 4regular networks are nonplanar, have a diameter growing as ~2^[\sqrt(\log_2 N^2)], exhibit "ballistic" diffusion (d_w=1), and a nontrivial ferromagnetic transition, T_c>0. It suggest that the 3regular networks are still quite "geometric", while the 4regular networks qualify as true smallworld networks with meanfield properties. As an example of an application we discuss synchronization of processors on these networks.
 Publication:

arXiv eprints
 Pub Date:
 December 2007
 arXiv:
 arXiv:0712.1259
 Bibcode:
 2007arXiv0712.1259B
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 9 pages, final version for JPA FastTrack, find related articles at http://www.physics.emory.edu/faculty/boettcher