### The average degree for vertices distributed uniformly

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6. [Random Geometric Graphs: Properties, Generation, Display]
(i) Given the number of vertices n and the target expectation d of the average degree for vertices distributed uniformly on the surface of the unit radius sphere, determine a formula for the threshold r such that a random geometric graph on a sphere Gs (n, r) should have average degree close to d.
(ii) Generate a Gs (n, r) for n = 2,000 with expected average degree d = 40. Using the plane sweep method count how many pairs are checked for the distance bound edge verification and compare with the number of all pairs.
(iii)Generate and display a sample Gs (n, r) restricted to a hemisphere with n = 500 in the hemisphere and expected average degree d = 12. Determine the maximum degree, average degree, and minimum degree of your sample and highlight a vertex and its incident edges for a vertex of minimum degree in your display.

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