Faculty of Mathematics and Mechanics, Saratov State University, Saratov 410012, Russian Federation.
Faculty of Computer Science and Information Technology, Saratov State University, Saratov 410012, Russian Federation.
Chaos. 2023 Jun 1;33(6). doi: 10.1063/5.0148803.
We examine the dynamics for the average degree of a node's neighbors in complex networks. It is a Markov stochastic process, and at each moment of time, this quantity takes on its values in accordance with some probability distribution. We are interested in some characteristics of this distribution: its expectation and its variance, as well as its coefficient of variation. First, we look at several real communities to understand how these values change over time in social networks. The empirical analysis of the behavior of these quantities for real networks shows that the coefficient of variation remains at high level as the network grows. This means that the standard deviation and the mean degree of the neighbors are comparable. Then, we examine the evolution of these three quantities over time for networks obtained as simulations of one of the well-known varieties of the Barabási-Albert model, the growth model with nonlinear preferential attachment (NPA) and a fixed number of attached links at each iteration. We analytically show that the coefficient of variation for the average degree of a node's neighbors tends to zero in such networks (albeit very slowly). Thus, we establish that the behavior of the average degree of neighbors in Barabási-Albert networks differs from its behavior in real networks. In this regard, we propose a model based on the NPA mechanism with the rule of random number of edges added at each iteration in which the dynamics of the average degree of neighbors is comparable to its dynamics in real networks.
我们研究了复杂网络中节点邻居的平均度数的动态。它是一个马尔可夫随机过程，在每个时间点，这个数量都根据某个概率分布取其值。我们对这个分布的一些特征感兴趣：它的期望和方差，以及它的变异系数。首先，我们研究了几个真实的社区，以了解在社交网络中这些值随时间的变化。对真实网络中这些数量的行为的实证分析表明，随着网络的增长，变异系数保持在较高水平。这意味着标准差和邻居的平均度数相当。然后，我们研究了作为一种著名的 Barabási-Albert 模型变体的模拟网络中这些数量随时间的演化，该模型具有非线性优先连接（NPA）和每次迭代时固定数量的附加链接。我们从理论上证明，在这种网络中（尽管非常缓慢），节点邻居的平均度数的变异系数趋于零。因此，我们确定了 Barabási-Albert 网络中邻居平均度数的行为与其在真实网络中的行为不同。在这方面，我们提出了一个基于 NPA 机制的模型，该模型在每次迭代时都有添加边的随机数量规则，其中邻居的平均度数的动态与真实网络中的动态相当。
