Ribeiro Andre S, Lloyd-Price Jason, Kesseli Juha, Häkkinen Antti, Yli-Harja Olli
Computational Systems Biology Research Group, Tampere University of Technology, Finland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Nov;78(5 Pt 2):056108. doi: 10.1103/PhysRevE.78.056108. Epub 2008 Nov 24.
Mutual information between the time series of two dynamical elements measures how well their activities are coordinated. In a network of interacting elements, the average mutual information over all pairs of elements I is a global measure of the correlation between the elements' dynamics. Local topological features in the network have been shown to affect I . Here we define a generalized clustering coefficient C_{p} and show that this quantity captures the effects of local structures on the global dynamics of networks. Using random Boolean networks (RBNs) as models of networks of interacting elements, we show that the variation of I ( I averaged over an ensemble of RBNs with the number of nodes N and average connectivity k ) with N and k is caused by the variation of C_{p} . Also, the variability of I between RBNs with equal N and k is due to their distinct values of C_{p} . Consequently, we propose a rewiring method to generate ensembles of BNs, from ordinary RBNs, with fixed values of C_{p} up to order 5, while maintaining in- and out-degree distributions. Using this methodology, the dependency of C_{p} on N and k and the variability of I for RBNs with equal N and k are shown to disappear in RBNs with C_{p} set to zero. The I of ensembles of RBNs with fixed, nonzero C_{p} values, also becomes almost independent of N and k . In addition, it is shown that C_{p} exhibits a power-law dependence on N in ordinary RBNs, suggesting that the C_{p} affects even relatively large networks. The method of generating networks with fixed C_{p} values is useful to generate networks with small N whose dynamics have the same properties as those of large scale networks, or to generate ensembles of networks with the same C_{p} as some specific network, and thus comparable dynamics. These results show how a system's dynamics is constrained by its local structure, suggesting that the local topology of biological networks might be shaped by selection, for example, towards optimizing the coordination between its components.
两个动态元素的时间序列之间的互信息衡量了它们的活动协调程度。在一个相互作用元素的网络中，所有元素对的平均互信息I是元素动态相关性的全局度量。网络中的局部拓扑特征已被证明会影响I。在这里，我们定义了一个广义聚类系数Cp，并表明这个量捕捉了局部结构对网络全局动态的影响。使用随机布尔网络（RBN）作为相互作用元素网络的模型，我们表明I（I是在具有节点数N和平均连通性k的RBN集合上平均得到的）随N和k的变化是由Cp的变化引起的。此外，具有相同N和k的RBN之间I的变异性是由于它们不同的Cp值。因此，我们提出了一种重新布线的方法，从普通的RBN生成具有固定Cp值（最高到5阶）的BN集合，同时保持入度和出度分布。使用这种方法，对于Cp设置为零的RBN，Cp对N和k的依赖性以及具有相同N和k的RBN的I的变异性都消失了。具有固定非零Cp值的RBN集合的I也几乎变得与N和k无关。此外，结果表明在普通RBN中Cp对N呈现幂律依赖性，这表明Cp甚至会影响相对较大的网络。生成具有固定Cp值的网络的方法对于生成具有小N且其动态与大规模网络具有相同性质的网络，或者生成与某些特定网络具有相同Cp从而具有可比动态的网络集合很有用。这些结果表明了系统的动态是如何受到其局部结构的约束，这表明生物网络的局部拓扑可能是通过选择形成的，例如，朝着优化其组件之间的协调方向。
