Topologic distance in the Lucena network

The Lucena network (LN) is the dual of a multifractal partition of the square. We analyze the relation between the typical topologic distance l and the number of vertices N of the LN. The multifractal partition has one parameter rho which controls the geometrical asymmetry of the multifractal. In the limit of rho -> 1 the blocks of the partition are squared, the connections amont the blocks are short range, the LN is more regular and the relation l proportional to root N is observed. For the limit rho -> 0 the blocks are strongly asymmetric, long range connections appear, and the topologic distance follows l proportional to (log N)(alpha), a weak small world phenomenon. For any network size we calculate analytically the size of the minimum distance l(min) (rho -> 0) and the maximal distance l(max) (rho -> 1). The distance in the weak small world regime is calculated using the number of vertices inside a radius of length l and taking into account the network average connectivity and the exponent alpha.