Percolation in fractal spatial networks with long-range interactions

We study the emergence of a giant component in a spatial network where the nodes form a fractal set, and the interaction between the nodes has a long-range power-law behavior. The nodes are positioned in the metric space using a Levy flight procedure, with an associated scale-invariant step probability density function, that is then followed by a process of connecting each pair of nodes with a probability that depends on the distance between them. Since the nodes are positioned sequentially, we are able to calculate the probability for an edge between any two nodes in terms of their indexes and to map the model to the problem of percolation in a one-dimensional lattice with long-range interactions. This allows the identification of the conditions for which a percolation transition is possible. The system is characterized by two control parameters which determine the fractal dimension of the nodes and the power law decrease of the probability of a bond with the distance between the nodes. The competition between these two parameters forms an intricate phase diagram, which describes when the system has a stable giant component, and when percolation transitions occur. Understanding the structure of this class of spatial networks is important when analyzing real systems, which are frequently heterogeneous and include long-range interactions.