A planar network between short and long range behaviour

•The Lucena Network is the dual of a multifractal partition of the square. The Projected Lucena Network PLN is the Lucena Network projected into the square lattice.•By construction the PLN is a planar network that is embedded in the plane.•The PLN follows the relation P(r)∝r−δP(r)∝r−δ for P(r)P(r) the probability of a node be connected to another node at distance rr.•The fractal dimension, dfdf, of the PLN depends on its internal asymmetry, we identify two regimes: short range, δ≃3.9δ≃3.9, and long range, δ≃3.2δ≃3.2.•For the short range limit df≃2df≃2, the dimension of the embedding. For long range df≃2.5df≃2.5.

We study the Projection of Lucena Network PLN: a planar network whose nodes coincide with the sites of the square lattice. The PLN has one internal parameter ρρ, for ρ→1ρ→1 the network resembles a symmetric regular lattice with almost none highly connected node while ρ→0ρ→0 is more asymmetric and there are several highly connected nodes. We estimate P(r)P(r), the probability of a node be connected to another node at distance rr. The PLN follows the fractal scaling P(r)∝r−δP(r)∝r−δ with 3<δ<43<δ<4 according to ρρ. For ρ→1ρ→1 we have δδ close to 44 which is a signature of short range interactions, while ρ→0ρ→0 shows δ→3δ→3 and has long range interactions. In addition, the fractal dimension, dfdf, of the PLN behaves as: for ρ→1ρ→1, df=2df=2, the dimension of the embedding space while for ρ→0ρ→0, dfdf increases and the PLN shares some similarities with a small world graph.