Eur. Phys. J. B (2016) 89: 120
https://doi.org/10.1140/epjb/e2016-60669-6

Regular Article

Topologic distance in the Lucena network

¹ Escola de Ciências e Tecnologia, Campus Central Universidade Federal do Rio Grande do Norte, 59078-970 Natal - RN, Brazil
² Departamento de Biofísica e Farmacologia, Centro de Biociências, Universidade Federal do Rio Grande do Norte, 59072-970, Natal - RN, Brazil

^a e-mail: gfcorso@gmail.com

Received: 12 August 2015
Received in final form: 20 November 2015
Published online: 12 May 2016

Abstract

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 ρ which controls the geometrical asymmetry of the multifractal. In the limit of ρ → 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 ∝ √N is observed. For the limit ρ → 0 the blocks are strongly asymmetric, long range connections appear, and the topologic distance follows l ∝ (log N)^α, a weak small world phenomenon. For any network size we calculate analytically the size of the minimum distance l_min (ρ → 0) and the maximal distance l_max (ρ → 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 α.