Explicit laws of large numbers for random nearest-neighbour type graphs
نویسنده
چکیده
We give laws of large numbers (in the Lp sense) for the total length of the k-nearest neighbours (directed) graph and the j-th nearest neighbour (directed) graph in Rd, d ∈ N, with power-weighted edges. We deduce a law of large numbers for the standard nearest neighbour (undirected) graph. We give the limiting constants, in the case of uniform random points in (0, 1)d, explicitly. Also, we give explicit laws of large numbers for the total power-weighted length of the Gabriel graph and two further graphs that are related to the standard nearest-neighbour graph: the on-line nearest-neighbour graph and the minimal directed spanning forest.
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