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.

Project supported by the Nature Science Foundation of Hebei Province, China (Grant Nos. F2016501023, F2017501041), Fundamental Research Funds for the Central Universities, China (Grant No. N172304030), and the National Natural Science Foundation of China (Grant No. 61402088).

Yang Qing-Lin,Wang Li-Fu,Li Huan et al.. A spectral coarse graining algorithm based on relative distance[J]. Acta Physica Sinica, 2019, 68(10):
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doi:10.7498/aps.68.20181848.