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

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a $k$th neighboring site of degree 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-law edge-length distribution of the form $P_>(l)\sim l^{-s}$ with the marginal exponent s=1. In all these networks, lengths of shortest paths grow as a power of the distance and random walk is super-diffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.