Recently the framework of multi-layer networks was proposed as a new model of complex networks, and was used in widespread applications in many fields, such as the cascading failure, the information spreading, the link prediction and the network synchronization. In multi-layer networks, the correlation and the coupling between two layers of network structures might exist, so it is a very significant issue how to detect the structural correlation and quantify the correlation between the two layers. In this study we summarize and propose methods to measure the structural correlation of double-layer networks in three levels. The first level is to detect the overall connection relationship of the whole double-layer network. The second level is to test the degree correlation characteristics between all nodes at different layers. At last, the third level is to look for the connection relationship between the rich nodes at different layers. Although the three kinds of correlations are all dependent on the network statistics, these statistics are all without units. Furthermore, the sizes and the structures of different networks see a great difference. Therefore, absolute numerical values of some statistics often are not important and we put forward a variety of null models for double-layer networks as a reference. Through the hypothesis testing methods, we can quantify the structure correlation in double-layer networks, and try to analyze the intrinsic mechanism of inducing this kind of structure correlation. Finally, we use an empirical double-layer network(the global language multi-layer network)to verify the effectiveness of our methodology. This methodology can be used to detect the complex coupling between the layers in an empirical double-layer network, and for the better understanding of multi-layer networks and for new applications based on the structure complexity of multi-layer networks.
CUI Liyan
,
XU Xiaoke
. Correlation detection of double-layer network based on null models[J]. Science & Technology Review, 2017
, 35(14)
: 63
-74
.
DOI: 10.3981/j.issn.1000-7857.2017.14.008
[1] Lee K M, Min B, Goh K I. Towards real-world complexity:An introduc tion to multiplex networks[J]. The European Physical Journal B, 2015, 88(2):1-20.
[2] Boccaletti S, Bianconi G, Criado R, et al. The structure and dynamics of multilayer networks[J]. Physics Reports, 2014, 544(1):1-122.
[3] Milo R, Shenorr S, Itzkovitz S, et al. Network motifs:Simple building blocks of complex networks[J]. Science, 2002, 298(5594):824-827.
[4] Milo R. Superfamilies of evolved and designed networks[J]. Science, 2004, 303(5663):1538-1542.
[5] Foster J G, Foster D V, Grassberger P, et al. Edge direction and the structure of networks[J]. Proceedings of the National Academy of Sci ences of the United States of America, 2010, 107(24):10815-10820.
[6] Colizza V, Flammini A, Serrano M A, et al. Detecting rich-club order ing in complex networks[J]. Nature Physics, 2006, 2(3):110-115.
[7] Mahadevan P, Hubble C, Krioukov D, et al. Orbis:Rescaling degree correlations to generate annotated internet topologies[J]. ACM SIG COMM Computer Communication Review, 2007, 37(4):325-336.
[8] Gjoka M, Kurant M, Markopoulou A. 2.5K-graphs:From sampling to generation[C]. IEEE International Conference on Computer Communica tions, Turin, Italy:2012
[9] 汪小帆, 李翔, 陈关荣. 复杂网络引论——模型、结构与动力学[M]. 北京:高等教育出版社, 2012. Wang Xiaofan, Li Xiang, Chen Guanrong. Introduction to complex net works:Models, structures and dynamics[M]. Beijing:Higher Education Press, 2012:217-218.
[10] 刘军. QAP:测量"关系" 之间关系的一种方法[J]. 社会, 2007, 27(4):164-174. Liu Jun. QAP:A unique method of measuring "Relationships" in rela tional data[J]. Society, 2007, 27(4):164-174.
[11] Fraser A M, Swinney H L. Independent coordinates for strange attrac tors from mutual information. Physical Review A, 1986, 33(2):1134-1140.
[12] Bianconi G. Statistical mechanics of multiplex networks:Entropy and overlap[J]. Physical Review E, 2013, 87(6):62806.
[13] Parshani R, Rozenblat C, Ietri D, et al. Inter-similarity between cou pled networks[J]. Europhysics letters, 2010, 92(6):2470-2484.
[14] Szell M, Lambiotte R, Thurner S. Multirelational organization of largescale social networks in an online world[J]. Proceedings of the Nation al Academy of Sciences of the United States of America, 2010, 107(31):13636-13641.
[15] Min B, Yi S D, Lee K M, et al. Network robustness of multiplex net works with interlayer degree correlations[J]. Physical Review E, 2014, 89(4):42811.
[16] 姚尊强, 尚可可, 许小可. 加权网络的常用统计量[J]. 上海理工大学学报, 2012, 34(1):18-26. Yao Zunqiang, Shang Keke, Xu Xiaoke. Fundamental statistics of weighted networks[J]. Journal of University of ShangHai for Science and Technology, 2012, 34(1):18-26.
[17] Opsahl T, Colizza V, Panzarasa P. Prominence and control:The weighted rich-club effect[J]. Physical Retters Letters, 2008, 101(16):168702.
[18] Zhou S, Mondragon R J. The rich-club phenomenon in the internet to pology[J]. IEEE Communications Letters, 2004, 8(3):180-182.
[19] Amaral L A N, Guimera R. Complex networks:Lies, damned lies and statistics[J]. Nature Physics, 2006, 2(2):75-76.
[20] Mahadevan P, Krioukov D, Fall K, et al. Systematic topology analysis and generation using degree correlations[J]. ACM Sigcomm Computer Communication Review, 2006, 36(4):135-146.
[21] Orsini C, Dankulov M M, Colomerdesimón P, et al. Quantifying ran domness in real networks[J]. Nature Communications, 2015, 6:8627.
[22] Schreiber T, Schmitz A. Surrogate time series[J]. Physica D:Nonlinear Phenomena, 1999, 142(3/4):346-382.
[23] Ronen S, Goncalves B, Hu K Z, et al. Links that speak:The global language network and its association with global fame[J]. Proceedings of the National Academy of Sciences of the United States of America, 2014,111(52):E5616-E5622.
