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 Wx and a type ψx,where(Wx) x∈...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 Wx and a type ψx,where(Wx) x∈Z d are i.i.d.random variables,and(ψx) 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 pxy = 1 exp(λφψ x ψ y WxWy /| x-y | α),where φij 's are entries from a type matrix Φ.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 λ WCC c ∈(0,∞) and λ SCC c ∈(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.展开更多
文摘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 Wx and a type ψx,where(Wx) x∈Z d are i.i.d.random variables,and(ψx) 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 pxy = 1 exp(λφψ x ψ y WxWy /| x-y | α),where φij 's are entries from a type matrix Φ.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 λ WCC c ∈(0,∞) and λ SCC c ∈(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.
