Percolation in heterogeneous spatial networks with long-range interactions
We study the emergence of a giant component in a spatial network where the distribution of the metric distances between the nodes is scale-invariant, 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, and is then followed by a process of connecting each pair of nodes with a probability function that depends on the distance between them. A natural way to analyze the system is to consider the total probability for an edge between steps in term of their indexes, by summing over their possible positions. By doing so, a correspondence is found between this model and a model of percolation in a one-dimensional lattice with long-range interactions, which allows the identification of the conditions for which a percolation transition is possible. We find that the emergence of a giant component and percolation transitions is determined by a complicated phase diagram, that exhibits a transition from weak long-range interactions to strong long-range interactions.