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To what extent can one "prune" the links in a wireless network while retaining (almost) the same connectivity achieved when each node is connected to all nodes within its communication radius? In a nutshell, if each node explores a portion of its neighborhood sufficient to reach ≈ log(1/ε) other nodes in expectation, w.h.p. all but a fraction ε of the network joins the same connected component. Each node can then "locally" choose to maintain (at most) 4 links, and the network still retains w.h.p. essentially the same level of connectivity.