Multi-Type Directed Scale-Free Percolation

In this paper, we study a long-range percolation model on the lattice ℤ d with multi-type vertices and directed edges. Each vertex x ∊ ℤ d is independently assigned a non-negative weight W x and a type ψ x , where (W x ) x∊ℤ d are i.i.d. random variables, and (ψ x ) x∊ℤ d are also i.i.d. Conditionally on weights and types, and given λ, α > 0, the edges are independent and the probability that there is a directed edge from x to y is given by p xy = 1 − exp(−λφ ψ x ψ y W x W y /|x−y| α ), where φ ij 's are entries from a type matrix Φ. We show that, when the tail of the distribution of W x is regularly varying with exponent τ − 1, the tails of the out/in-degree distributions are both regularly varying with exponent γ = α(τ − 1)/d. We formulate conditions under which there exist critical values λ c WCC ∊ (0, ∞) and λ c SCC ∊ (0, ∞) such that an infinite weak component and an infinite strong component emerge, respectively, when λ exceeds them. A phase transition is established for the shortest path lengths of directed and undirected edges in the infinite component at the point γ = 2, where the out/in-degrees switch from having finite to infinite variances. The random graph model studied here features some structures of multi-type vertices and directed edges which appear naturally in many real-world networks, such as the SNS networks and computer communication networks.