A universal feature of open complex systems such as reaction networks in living organisms, ecosystems, and social communities is that their complexity emerges, or at least persists, under successive introductions of new elements. To have a general and simple understanding for the basic condition to let such systems grow, we investigate a simple mathematical process. It is found that the model system either grows infinitely large or stays finite depending upon the model’s unique parameter m, the average number of interactions per element. Comparing to the classical diversity-stability relation based on a linear-stability analysis, the condition for our system to grow is more moderate. The mechanism of this novel universal transition is described in detail. The characteristics of the model, such as lifetime distribution of the elements, and its relevance to the previous works such as SOC models are also discussed.
Phase transition Robustness and fragility Stability and diversity Biological systems Ecosystems Social systems Complex networks
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The author thanks H. Ando, C.-K. Hu, H. Kori, Y. Murase, T. Ohira, Y. Sano, S. Shimada, and C.-P. Zhu for discussion and comments. This work was supported by the JSPS Grant-in-Aid for Young Scientists (B) no. 21740284 MEXT, Japan. The systematic simulations in this study were assisted by OACIS .
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