利用行列式计算自相似网络的生成树
Counting spanning trees in self-similar networks by evaluating determinants
摘要：
生成树与网络的各个方面相关。通常，网络中的生成树的数量可以通过计算网络的拉普拉斯矩阵的相关行列式来获得。然而，对于大型通用网络，评估相关行列式在计算上是难以处理的。在本文中，我们开发了一个相当通用的技术，用于计算自相似网络的行列式，从而提供了一种方法来确定网络中表现出自相似性的生成树的数量。我们描述了一个家庭的网络，称为$(x,y)$ -花，显示丰富的行为，在各种各样的真实系统中观察到的计算过程。生成树的枚举基于不同代的$(x,y)$ -花对应的拉普拉斯矩阵的子矩阵的行列式之间的关系，无需直接费力地计算行列式。利用所提出的方法，我们得到了解析的确切数目的生成树的$(x,y)$ -花，在此基础上，我们还得到了这些网络中的生成树的熵。此外，为了说明我们的技术的通用性，我们将其应用到其他一些具有不同的度分布的自相似网络，并获得显式的解决方案的数量的生成树和它们的熵。最后，我们比较我们的结果具有相同的平均度，但不同的结构属性，如度分布和分形维数的网络，并揭示这些拓扑特征的生成树的数量的影响。
英文摘要：
Spanning trees are relevant to various aspects of networks. Generally, the number of spanning trees in a network can be obtained by computing a related determinant of the Laplacian matrix of the network. However, for a large generic network, evaluating the relevant determinant is computationally intractable. In this paper, we develop a fairly generic technique for computing determinants corresponding to self-similar networks, thereby providing a method to determine the numbers of spanning trees in networks exhibiting self-similarity. We describe the computation process with a family of networks, called $(x,y)$-flowers, which display rich behavior as observed in a large variety of real systems. The enumeration of spanning trees is based on the relationship between the determinants of submatrices of the Laplacian matrix corresponding to the $(x,y)$-flowers at different generations and is devoid of the direct laborious computation of determinants. Using the proposed method, we derive analytically the exact number of spanning trees in the $(x,y)$-flowers, on the basis of which we also obtain the entropies of the spanning trees in these networks. Moreover, to illustrate the universality of our technique, we apply it to some other self-similar networks with distinct degree distributions, and obtain explicit solutions to the numbers of spanning trees and their entropies. Finally, we compare our results for networks with the same average degree but different structural properties, such as degree distribution and fractal dimension, and uncover the effect of these topological features on the number of spanning trees.
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