The maximum and minimum degrees of random bipartite multigraphs

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows: let m = m(n) be a positive integer-valued function on n   and GG(n, m; {pk}) the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A = {a1, a2, … an} and B = {b1, b2 … bm}, in which the numbers tai,bjtai,bj of the edges between any two vertices ai ∈ A and bj ∈ B are identically distributed independent random variables with distributionP{tai,bj = k} = pk, k = 0, 1, 2, ⋯,P{tai,bj = k} = pk, k = 0, 1, 2, ⋯,where pk ≥ 0 and ∑pk = 1k=0∞. They obtain that Xc,d,A, the number of vertices in A with degree between c and d of Gn, m ∈ GG (n, m;{pk}) has asymptotically Poisson distribution, and answer the following two questions about the space GG(n, m;{pk}) with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D(n) such that almost every random multigraph Gn, m ∈ GG (n, m;{pk}) has maximum degree D(n) in A? under which condition for {pk} has almost every multigraph Gn, m ∈ GG (n, m;{pk}) a unique vertex of maximum degree in A?