  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.
