Amari Shun-ichi, Ando Hiroyasu, Toyoizumi Taro, Masuda Naoki
RIKEN Brain Science Institute, Hirosawa 2-1, Wako, Saitama 351-0198, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Feb;87(2):022814. doi: 10.1103/PhysRevE.87.022814. Epub 2013 Feb 21.
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.
我们研究了由二元布尔元素组成的随机连接网络以及由二元多数投票元素组成的随机连接网络的动力学。我们阐明了它们在稀疏连接和密集连接情况下的差异。大型网络动力学的快速性通常通过瞬态路径的长度来量化，这是一种难以进行解析处理的度量。对于二元元素网络的离散时间动力学，我们通过使用一个称为状态集中度的概念，在一个替代的统一框架中解决了这一困境，状态集中度被定义为状态转移图中t步祖先平均数量的指数。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接定义。利用这个指数，我们研究了随机布尔网络和多数投票网络的动力学。我们发现，极其稀疏的布尔网络和任意密度的多数投票网络都能实现快速性，部分原因是入度分布呈长尾状。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。
