Superdiffusion in a class of networks with marginal long-range connections

摘要

A class of cubic networks composed of a regular one-dimensional lattice and aset of long-range links is introduced. Networks parametrized by a positiveinteger k are constructed by starting from a one-dimensional lattice anditeratively connecting each site of degree 2 with a $k$th neighboring site ofdegree 2. Specifying the way pairs of sites to be connected are selected,various random and regular networks are defined, all of which have a power-lawedge-length distribution of the form $P_>(l)\sim l^{-s}$ with the marginalexponent s=1. In all these networks, lengths of shortest paths grow as a powerof the distance and random walk is super-diffusive. Applying a renormalizationgroup method, the corresponding shortest-path dimensions and random-walkdimensions are calculated exactly for k=1 networks and for k=2 regularnetworks; in other cases, they are estimated by numerical methods. Although,s=1 holds for all representatives of this class, the above quantities are foundto depend on the details of the structure of networks controlled by k and otherparameters.