Masuda Naoki, Kawamura Yoji, Kori Hiroshi
Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Oct;80(4 Pt 2):046114. doi: 10.1103/PhysRevE.80.046114. Epub 2009 Oct 19.
Many complex networks are described by directed links; in such networks, a link represents, for example, the control of one node over the other node or unidirectional information flows. Some centrality measures are used to determine the relative importance of nodes specifically in directed networks. We analyze such a centrality measure called the influence. The influence represents the importance of nodes in various dynamics such as synchronization, evolutionary dynamics, random walk, and social dynamics. We analytically calculate the influence in various networks, including directed multipartite networks and a directed version of the Watts-Strogatz small-world network. The global properties of networks such as hierarchy and position of shortcuts rather than local properties of the nodes, such as the degree, are shown to be the chief determinants of the influence of nodes in many cases. The developed method is also applicable to the calculation of the PAGERANK. We also numerically show that in a coupled oscillator system, the threshold for entrainment by a pacemaker is low when the pacemaker is placed on influential nodes. For a type of random network, the analytically derived threshold is approximately equal to the inverse of the influence. We numerically show that this relationship also holds true in a random scale-free network and a neural network.
许多复杂网络是由有向链接描述的；在这样的网络中，例如，一个链接表示一个节点对另一个节点的控制或单向信息流。一些中心性度量用于确定节点在有向网络中的相对重要性。我们分析一种称为影响力的中心性度量。影响力表示节点在各种动态过程中的重要性，如同步、进化动态、随机游走和社会动态。我们通过分析计算各种网络中的影响力，包括有向多部网络和Watts-Strogatz小世界网络的有向版本。在许多情况下，网络的全局属性，如层次结构和捷径的位置，而不是节点的局部属性，如度，被证明是节点影响力的主要决定因素。所开发的方法也适用于计算网页排名。我们还通过数值模拟表明，在耦合振荡器系统中，当起搏器放置在有影响力的节点上时，起搏器同步的阈值较低。对于一种随机网络类型，通过分析得出的阈值大约等于影响力的倒数。我们通过数值模拟表明，这种关系在随机无标度网络和神经网络中也成立。
