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My copy of the book being in my locked-down office makes it easy to avoid checking which section this question is from for a hint of the expected method, so here's a sledge hammer. Fix a partition $V = V_1 \cup V_2 \cup \cdots \cup V_{1000}$ into independent sets and let $U_i = U \cap V_i$. A uniformly random subset of $V$ includes each element of $V$ ...


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For $p= \frac c N$, the mean number of components per node is $(1-s_0) (1- \frac c 2 (1-s_0))$, where $s_0$ is the fraction of nodes in the largest component. (So, trivially, in the non-percolating phase for $c<1$, where $s_0=0$, one obtains $1- \frac c 2$, which corresponds to the simple argument to having $N$ nodes and $M=\frac {N(N-1)} 2 \frac c N= \...


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The wikipedia article on random geometric graphs only scratches the surface. A much deeper treatment is provided in Mathew Penrose's amazing text Random Geometric Graphs. Chapter 4 contains a treatment of what you are asking about, namely the "empirical distribution of nearest-neighbour distances amongst the points." Also, the underlying ...


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