SHANG Yi-Lun
理论物理通讯. 2012, 57(4): 701-716.
In this paper, we study a long-range percolation model on the lattice Zd with multi-type vertices and directed edges. Each vertex x∈Zd is independently assigned a non-negative weight Wx and a type Ψx, where (Wx)_x∈Zd are i.i.d.\ random variables, and (Ψx)_x∈Zd 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 pxy =1-\exp(-λφΨxψy WxWy/|x-y|α), where φij's are entries from a type matrix ψ. We show that, when the tail of the distribution of Wx 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 λcWCC ∈(0,∞) and λcSCC ∈(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.
