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_Ψxψy W_xW_y/|x-y|^α), where φ_ij's are entries from a type matrix ψ. 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.