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Accurately and quickly calculating the weighted spectral distribution

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Abstract

The weighted spectral distribution (WSD) is a measure defined on the normalized Laplacian spectrum. It can be used for comparing complex networks with different sizes (number of nodes) and provides a sensitive discrimination of the structural robustness of complex networks. In this paper, we design an algorithm for the accurate calculation of the WSD in large-scale complex networks by utilizing characteristics of the graph structure. As an extension to Sylvester’s Law of Inertia for the calculation of the WSD based on spectral characteristics, our algorithm exhibits a higher time efficiency when applied to large-scale complex networks.

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Acknowledgments

We would like to thank the anonymous reviewers for their comments and suggestions that helped improve this paper. This paper is supported by the National Natural Science Foundation of China with Grant Nos. 61402485, 71201169 and 61303061.

Author information

Authors and Affiliations

  1. Luoyang Electronic Equipment Test Center, Luoyang, 471003, China

    Bo Jiao, Gang Lu, Ying Zhou & Jing Du

  2. College of Computer, National University of Defense Technology, Changsha, 410073, China

    Yuan-ping Nie

  3. College of Information Systems and Management, National University of Defense Technology, Changsha, 410073, China

    Jian-mai Shi

Authors
  1. Bo Jiao
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  2. Yuan-ping Nie
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  3. Jian-mai Shi
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  4. Gang Lu
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  5. Ying Zhou
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  6. Jing Du
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Corresponding author

Correspondence to Bo Jiao.

Appendix

Appendix

In this paper, we focus on the calculation of the WSD with \(N = 4\), and the reasons for selecting the parameter N are analyzed in Sect. 2.2. However, the WSD with other values for N can be studied. Fay and Haddadi et al. [6] considered that \(N = 3\) is related to the well-known clustering coefficient and that \(N = 4\) as a 4-cycle represents two routes (i.e., minimal redundancy) between two nodes. In addition, they indicated that other values of N may be of interest for other applications [6]. Now, we use the methods in Sect. 3 to find all patterns of 5-cycles with start node A, as shown in Fig. 10. Moreover, our methods for calculating the WSD with \(N = 4\) can be extended to higher numbers of cycles. Due to space limitations, the algorithms with other values for N are not analyzed in detail in this paper and will be studied in our future work.

Fig. 10
figure 10

All patterns of 5-cycles with start node A

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Jiao, B., Nie, Yp., Shi, Jm. et al. Accurately and quickly calculating the weighted spectral distribution. Telecommun Syst 62, 231–243 (2016). https://doi.org/10.1007/s11235-015-0077-7

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