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Objective  The regularity of scale-free patterns in rank-size relations has been observed in word frequency, city size distribution, firm size distribution, and gene expression. Because of the common emergence of this regularity, understanding its mechanisms has been of great interest. For obtaining the scale-free pattern regularity, various models based on the rich-get-richer mechanism have been proposed; however, the overarching procedure of searching for the “rich” is in disagreement with the locally interacting behaviors seen in the aforementioned natural and social phenomena.    Results  We implemented a computational model of a resource distribution system inspired by observations of word connectivity, which is created by local constraints with periodic and phase modulatory features. Here, we empirically demonstrated that a phase-modulated periodic connecting system can reach a dynamic equilibrium state as the most probable case, with the self-organizing scale-free patterns. The regularity could be a result of the configurational balance in spatiotemporal inequity during the resource distribution process with an adaptive constrained connectivity. Our results suggest that investigations of interferences of oscillating fluctuations in the system will elucidate the autoregulatory dynamic behavior.    Electronic supplementary material  The online version of this article (10.1186/s13104-019-4149-8) contains supplementary material, which is available to authorized users.

Self-organizing scale-free patterns in a phase-modulated periodic connecting system