where 𝑘 𝑚𝑖𝑛 is the number of edges added to form a minimally connected design. The sum in equation 11 starts at an index of 2 because 𝑘 𝑐 is the number of edges to transition from 𝑐 → 𝑐 -1 and 1 is the lower bound for 𝑐. For a geometric distribution such as the one in Eq. 10, the expectation value for 𝑘 𝑐 is the inverse of the probability ) that adding an edge decreases 𝑐. By linearity of expectation, it was then previously shown that