Adding o(n²) random edges to an αn-regular graph yields the same asymptotic minimum weights for spanning trees, shortest paths, and bipartite matchings as in the complete graph, scaled by α^{-1}.
Aldous, Asymptotics in the random assignment problem, Probability Theory and Related Fields 93 (1992) 507-534
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The effect of adding randomly weighted edges
Adding o(n²) random edges to an αn-regular graph yields the same asymptotic minimum weights for spanning trees, shortest paths, and bipartite matchings as in the complete graph, scaled by α^{-1}.