Kauffman型网络中状态集中指数的快速性度量
State Concentration Exponent as a Measure of Quickness in Kauffman-type Networks
摘要：
我们研究了由二元布尔元素和二元多数表决元素组成的随机连接网络的动力学。我们阐明他们的差异，在稀疏和密集连接的情况下。大型网络动态的快速性通常用瞬态路径的长度来量化，这是一个难以分析的度量。对于离散时间动态的二进制元素的网络，我们解决这个困境的一个替代的统一框架，通过使用一个概念，称为状态浓度，定义为指数的平均数的t步祖先在状态转换图。状态转移图由对应于网络状态的节点和对应于转移的有向链路定义。使用这个指数，我们询问随机布尔和多数投票网络的动态。我们发现，具有任意密度的极稀疏布尔网络和多数投票网络实现了快速性，部分原因是长尾入度分布。作为一个推论，只有相对密集的多数投票网络才能同时实现快速性和鲁棒性。
英文摘要：
We study the dynamics of randomly connected networks composed of binary Boolean elements and those composed of binary majority vote elements. We elucidate their differences in both sparsely and densely connected cases. The quickness of large network dynamics is usually quantified by the length of transient paths, an analytically intractable measure. For discrete-time dynamics of networks of binary elements, we address this dilemma with an alternative unified framework by using a concept termed state concentration, defined as the exponent of the average number of t-step ancestors in state transition graphs. The state transition graph is defined by nodes corresponding to network states and directed links corresponding to transitions. Using this exponent, we interrogate the dynamics of random Boolean and majority vote networks. We find that extremely sparse Boolean networks and majority vote networks with arbitrary density achieve quickness, owing in part to long-tailed in-degree distributions. As a corollary, only relatively dense majority vote networks can achieve both quickness and robustness.
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