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I place $N$ points on a circular plane of radius $R$, and draw edges to connect points that are less than or equal to some distance $D$ to form a set of graphs or cliques $G_i$. As a function of $N$, $R$, and $D$, what is the mean number of points, $m_k$, that participate in graphs with $k$ or fewer vertices?

To clarify, the $N$ points are randomly placed on the circular plane with uniform probability across its surface.

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I've rephrased your title to simplify it a bit - see . Feel free to revert it if I have misread your question. – j.c. Feb 15 '12 at 16:04
@jc Thanks for the new title suggestion, I like it. – EclipseInterlude Feb 15 '12 at 16:09

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