APPENDICES to “Capacity Scaling of Wireless Social Networks”

APPENDIX A USEFUL KNOWN RESULTS Lemma A.1 (Minimal Spanning Tree, [1]): Let Xi, 1 ≤ i < ∞, denote independent random variables with values in R, d ≥ 2, and let Mn denote the cost of a minimal spanning tree of a complete graph with vertex set {Xi}i=1, where the cost of an edge (Xi, Xj) is given by Ψ(|Xi − Xj |). Here, |Xi−Xj | denotes the Euclidean distance between Xi and Xj and Ψ is a monotone function. For bounded random variables and 0 < σ < d, it holds that as n → ∞, with probability 1, one has Mn ∼ c1(σ, d) · n d−σ d · ∫ Rd f(X) d−σ d dX , provided Ψ(x) ∼ x , where f(X) is the density of the absolutely continuous part of the distribution of the {Xi}. Lemma A.2 (Kolmogorov’s Strong LLN, [2]): Let {Xn} be an i.i.d. sequence of random variables having finite mean: for ∀n, E[Xn] < ∞. Then, a strong law of large numbers (LLN) applies to the sample mean: X̄n a.s. −→ E[Xn], where a.s. −→ denotes almost sure convergence.