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I am currently faced with two similar problems.

First: given a random graph $G(n,p)$ in the Erdös-Renyi-model and a randomly chosen edge $e$, what is the expected number of shortest paths using $e$?

Second: given a complete graph with edge weights uniformly randomly chosen in the interval $[1, 1000]$ and a random edge of weight $\ell$. What is the expected number of shortest paths in the graph using the selected edge?

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closed as off-topic by Matthew Conroy, JonMark Perry, Claude Leibovici, Adriano, erfink May 17 '17 at 8:19

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