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We study the class of networks which can be created in polylogarithmicparallel time by network constructors: groups of anonymous agents that interactrandomly under a uniform random scheduler with the ability to form connectionsbetween each other. Starting from an empty network, the goal is to construct astable network which belongs to a given family. We prove that the class oftrees where each node has any k >= 2 children can be constructed in O(log n)parallel time with high probability. We show that constructing networks whichare k-regular is Omega(n) time, but a minimal relaxation to (l, k)-regularnetworks, where l = k - 1 can be constructed in polylogarithmic parallel timefor any fixed k, where k > 2. We further demonstrate that when the finite-stateassumption is relaxed and k is allowed to grow with n, then k = log log n actsas a threshold above which network construction is again polynomial time. Weuse this to provide a partial characterisation of the class of polylogarithmictime network constructors.

On the Distributed Construction of Stable Networks in Polylogarithmic Parallel Time