Statistical Mechanics of Steiner trees
  The Minimum Weight Steiner Tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows to analyze the statistical mechanics properties of MST on random graphs of various types. 
M. Bayati, C. Borgs, A. Braunstein, J. Chayes, A. Ramezanpour, R.   Zecchina