基于相对距离的复杂网络谱粗粒化方法

复杂网络的同步作为一种重要的网络动态特性,在通信、控制、生物等领域起着重要的作用.谱粗粒化方法是一种在保持原始网络的同步能力尽量不变情况下将大规模网络约简为小规模网络的算法.此方法在对约简节点分类时是以每个节点对应特征向量分量间的绝对距离作为判断标准,在实际运算中计算量大,可执行性较差.本文提出了一种以特征向量分量间相对距离作为分类标准的谱粗粒化改进算法,能够使节点的合并更加合理,从而更好地保持原始网络的同步能力.通过经典的三种网络模型(BA无标度网络、ER随机网络、NW小世界网络)和27种不同类型实际网络的数值仿真分析表明,本文提出的算法对比原来的算法能够明显改善网络的粗粒化效果,并发现互联网、生物、社交、合作等具有明显聚类结构的网络在采用谱粗粒化算法约简后保持同步的能力要优于电力、化学等模糊聚类结构的网络.

A spectral coarse graining algorithm based on relative distance

As a key approach to understanding complex systems (e.g. biological, physical, technological and social systems), the complex networks are ubiquitous in the whole world. Synchronization in complex networks is significant for a more in-depth understanding of the dynamic characteristics of the networks, where tremendous efforts have been devoted to their mechanism and applications in the last two decades. However, many real-world networks consist of hundreds of millions of nodes. Studying the synchronization of such large-scale complex networks often requires solving a huge number of coupled differential equations, which brings great difficulties to both computation and simulation. Recently, a spectral coarse graining approach was proposed to reduce the large-scale network into a smaller one while maintaining the synchronizability of the original network. The absolute distance between the eigenvector components corresponding to the minimum non-zero eigenvalues of the Laplacian matrix is used as a criterion for classifying the nodes without considering the influence of the relative distance between eigenvector components in an original spectral coarse graining method. By analyzing the mechanism of the spectral coarse graining procedure in preserving the synchronizability of complex networks, we prove that the ability of spectral coarse graining to preserve the network synchronizability is related to the relative distance of the eigenvector components corresponding to the merged nodes. Therefore, the original spectral coarse graining algorithm is not satisfactory enough in node clustering. In this paper, we propose an improved spectral coarse graining algorithm based on the relative distance between eigenvector components, in which we consider the relative distance between the components of eigenvectors for the eigenvalues of network coupling matrix while clustering the same or similar nodes in the network, thereby improving the clustering accuracy and maintaining the better synchronizability of the original network. Finally, numerical experiments on networks of ER random, BA scale-free, WS small-world and 27 different types of real-world networks are provided to demonstrate that the proposed algorithm can significantly improve the coarse graining effect of the network compared with the original algorithm. Furthermore, it is found that the networks with obvious clustering structure such as internet, biological, social and cooperative networks have better ability to maintain synchronization after reducing scale by spectral coarse-grained algorithm than the networks of fuzzy clustering structure such as power and chemical networks.