Dorogovtsev S N, Mendes J F
Departamento de Física and Centro de Física do Porto, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 May;63(5 Pt 2):056125. doi: 10.1103/PhysRevE.63.056125. Epub 2001 Apr 26.
The scaling behavior of scale-free evolving networks, arising in areas such as communications, scientific citations, collaborations, etc., is studied. We derive universal scaling relations describing properties of such networks, and indicate the limits of their validity. We show that the main properties of scale-free evolving networks may be described in the framework of a simple continuous approach. The simplest models of networks, growing according to a mechanism of preferential attachment of links to nodes, are used. We consider different forms of this preference, and demonstrate that the range of preferential attachments producing scale-free networks is wide. We also obtain scaling relations for networks with nonlinear, accelerating growth, and describe the temporal evolution of the arising distributions. Size effects-the cutoffs of these distributions-introduce restrictions for the observation of power-law dependences. Mainly we discuss the so-called degree distribution, i.e., the distribution of the number of connections of nodes. A scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied. We describe the effects of differences between nodes. The "aging" of nodes changes the exponents of the distributions. The appearance of a single node with high fitness changes the degree distribution of a network dramatically. If its fitness exceeds some threshold value, this node captures a finite part of all links of the network. We show that permanent random damage to a growing scale-free network-a permanent deletion of some links-radically changes the values of the scaling exponents. Results of other kinds of permanent damage are described.
研究了在通信、科学引文、合作等领域出现的无标度演化网络的标度行为。我们推导出描述此类网络性质的通用标度关系，并指出其有效性的限制。我们表明，无标度演化网络的主要性质可以在一种简单的连续方法框架内进行描述。使用了根据链接到节点的优先连接机制生长的最简单网络模型。我们考虑了这种偏好的不同形式，并证明产生无标度网络的优先连接范围很广。我们还获得了具有非线性加速增长的网络的标度关系，并描述了所产生分布的时间演化。尺寸效应——这些分布的截断——对幂律依赖性的观测引入了限制。主要讨论所谓的度分布，即节点连接数的分布。还研究了不断增长的引文网络中单个节点对之间链接分布的标度形式。我们描述了节点差异的影响。节点的“老化”会改变分布的指数。单个具有高适应性节点的出现会极大地改变网络的度分布。如果其适应性超过某个阈值，该节点会捕获网络所有链接的有限部分。我们表明，对不断增长的无标度网络进行永久性随机破坏——永久性删除一些链接——会从根本上改变标度指数的值。描述了其他类型永久性破坏的结果。
