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Given $n$ samples of a regular discrete distribution $\pi$, we prove in thisarticle first a serial of SLLNs results (of Dvoretzky and Erd\"{o}s' type)which implies a typical power law when $\pi$ is heavy-tailed. Constructing a(random) graph from the ordered $n$ samples, we can establish other laws forthe degree-distribution of the graph. The phenomena of small world is alsodiscussed.

Range-Renewal Processes: SLLN, Power Law and Beyonds