Let G = G n,k denote the graph formed by placing points in a square of area n according to a Poisson process of density 1 and joining each point to its k nearest neighbours. In [2] Balister, Bollobás, Sarkar and Walters proved that if k < 0.3043 log n then the probability that G is connected tends to 0, whereas if k > 0.5139 log n then the probability that G is connected tends to 1. We prove that, around the threshold for connectivity, all vertices near the boundary of the square are part of the (unique) giant component. This shows that arguments about the connectivity of G do not need to consider 'boundary' effects. We also improve the upper bound for the threshold for connectivity of G to k = 0.4125 log n.
Reference REF improves the upper bound to be 0.4125 log .
Small components in k-nearest neighbour graphs
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Computer Science Mathematics