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摘要翻译：
本文对任意度分布的无向随机布尔网络中的临界线进行了分析和数值研究，包括连接的无标度拓扑$p(k)\sim k^{-\gamma}$。我们证明了在无限大的无标度网络中，冻结相与混沌相的转变发生在$3<γ<3.5$。这一观察之所以有趣，有两个原因。首先，由于无标度网络中的大多数临界现象在$\γ<3$时都表现出非平凡的性质，因此Kauffman模型中临界线的位置似乎是一个重要的例外。第二，由于基因调控网络具有$γ<3$的无标度拓扑结构，在有限规模的网络中，上述过渡向更小的$γ$转移的观察是Kauffman模型作为模拟真实系统的良好起点的一个论据。我们还解释了经典随机图数值模拟中临界线的不可达性是由于渗流现象所致。
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英文标题：
《Kauffman Boolean model in undirected scale free networks》
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作者：
Piotr Fronczak, Agata Fronczak and Janusz A. Holyst
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最新提交年份：
2007
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分类信息：
一级分类：Physics 物理学
二级分类：Disordered Systems and Neural Networks 无序系统与神经网络
分类描述：Glasses and spin glasses; properties of random, aperiodic and quasiperiodic systems; transport in disordered media; localization; phenomena mediated by defects and disorder; neural networks
眼镜和旋转眼镜；随机、非周期和准周期系统的性质；无序介质中的传输；本地化；由缺陷和无序介导的现象；神经网络
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一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
We investigate analytically and numerically the critical line in undirected random Boolean networks with arbitrary degree distributions, including scale-free topology of connections $P(k)\sim k^{-\gamma}$. We show that in infinite scale-free networks the transition between frozen and chaotic phase occurs for $3<\gamma < 3.5$. The observation is interesting for two reasons. First, since most of critical phenomena in scale-free networks reveal their non-trivial character for $\gamma<3$, the position of the critical line in Kauffman model seems to be an important exception from the rule. Second, since gene regulatory networks are characterized by scale-free topology with $\gamma<3$, the observation that in finite-size networks the mentioned transition moves towards smaller $\gamma$ is an argument for Kauffman model as a good starting point to model real systems. We also explain that the unattainability of the critical line in numerical simulations of classical random graphs is due to percolation phenomena.
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PDF链接：
https://arxiv.org/pdf/707.1963
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