Given a complete symmetric weighted graph with $n$ vertices, for such a graph there always exists a minimum spanning tree and, under the assumption of the uniqueness of that tree, the vertex degrees will have a specific (discrete) distribution.
Question
for which statistical properties of non-trivial distributions of edgelengths, per vertex and per the entire graph, do the MSTs of random complete symmetric graphs with those edge-lengths have the following property:
the distribution of vertex degrees in "secondary" MSTs induced by the vertices of degree $k$ in the "primary" MST is the same as in the primary MST
