随机布尔网络中的噪声
Noise in random Boolean networks
摘要：
我们研究了噪声对随机布尔网络的影响。噪声被实现为节点不遵守其确定性更新规则的概率$p$。我们定义了两个序参量，即有噪声和无噪声网络之间的汉明距离的长时间平均值，以及平均冻结度，它是衡量节点偏好两个布尔状态之一的程度的一个指标。我们评估这两个序参量作为噪声强度的函数，找到一个平稳的过渡，从确定性（$p=0$）完全随机（$p=1/2$）动态网络与$K\le2$，和一个一阶过渡在$p=0$为$K>2$。计算机模拟得到的结果大多也是解析式的。平均汉明距离可以使用退火近似来计算。为了获得作为噪声强度函数的冻结度分布，必须进行更复杂的自洽计算。该分布是K=1的δ峰的集合，并且它具有$K>1$的分形结构，在极限$K\gg1$中接近连续分布。
英文摘要：
We investigate the effect of noise on Random Boolean Networks. Noise is implemented as a probability $p$ that a node does not obey its deterministic update rule. We define two order parameters, the long-time average of the Hamming distance between a network with and without noise, and the average frozenness, which is a measure of the extent to which a node prefers one of the two Boolean states. We evaluate both order parameters as function of the noise strength, finding a smooth transition from deterministic ($p=0$) to fully stochastic ($p=1/2$) dynamics for networks with $K\le2$, and a first order transition at $p=0$ for $K>2$. Most of the results obtained by computer simulation are also derived analytically. The average Hamming distance can be evaluated using the annealed approximation. In order to obtain the distribution of frozenness as function of the noise strength, more sophisticated self-consistent calculations had to be performed. This distribution is a collection of delta peaks for K=1, and it has a fractal sructure for $K>1$, approaching a continuous distribution in the limit $K\gg1$.
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