On the value of a random minimum weight steiner tree
Consider a complete graph on n vertices with edge weights chosen randomly and independently from an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1 + o(1))(k - 1)(logn - log k)/n when k = o(n) and n → ∞.
Bollobás, B., Gamarnik, D., Riordan, O., & Sudakov, B. (2004). On the value of a random minimum weight steiner tree. Combinatorica, 24 (2), 187-207. https://doi.org/10.1007/s00493-004-0013-z