×

Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions. (English) Zbl 1456.82634

Summary: Many studies have shown that additional information can be gained on time series by investigating their associated complex networks. In this work, we investigate the multifractal property and Laplace spectrum of the horizontal visibility graphs (HVGs) constructed from fractional Brownian motions. We aim to identify via simulation and curve fitting the form of these properties in terms of the Hurst index \(H\). First, we use the sandbox algorithm to study the multifractality of these HVGs. It is found that multifractality exists in these HVGs. We find that the average fractal dimension \(\langle D(0)\rangle\) of HVGs approximately satisfies the prominent linear formula \(\langle D(0)\rangle =2-H\); while the average information dimension \(\langle D(1)\rangle\) and average correlation dimension \(\langle D(2)\rangle\) are all approximately bi-linear functions of \(H\) when \(H\geqslant 0.15\). Then, we calculate the spectrum and energy for the general Laplacian operator and normalized Laplacian operator of these HVGs. We find that, for the general Laplacian operator, the average logarithm of second-smallest eigenvalue \(\langle \ln \left({{u}_2}\right)\rangle \), the average logarithm of third-smallest eigenvalue \(\langle \ln \left({{u}_3}\right)\rangle \), and the average logarithm of maximum eigenvalue \(\langle \ln \left({{u}_n}\right)\rangle\) of these HVGs are approximately linear functions of \(H\); while the average Laplacian energy \(\langle{{E}_{\text{nL}}}\rangle\) is approximately a quadratic polynomial function of \(H\). For the normalized Laplacian operator, \( \langle \ln \left({{u}_2}\right)\rangle\) and \(\langle \ln \left({{u}_3}\right)\rangle\) of these HVGs approximately satisfy linear functions of \(H\); while \(\langle \ln \left({{u}_n}\right)\rangle\) and \(\langle{{E}_{\text{nL}}}\rangle\) are approximately a 4th and cubic polynomial function of \(H\) respectively.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics

Software:

Boost; MatlabBGL
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albert R and Barabási A L 2002 Rev. Mod. Phys.74 47 · Zbl 1205.82086 · doi:10.1103/RevModPhys.74.47
[2] Song C, Havlin S and Makse H A 2005 Nature433 392 · doi:10.1038/nature03248
[3] Zhang J and Small M 2006 Phys. Rev. Lett.96 238701 · doi:10.1103/PhysRevLett.96.238701
[4] Xu X-K, Zhang J and Small M 2008 Proc. Natl Acad. Sci. USA105 19601 · Zbl 1202.37118 · doi:10.1073/pnas.0806082105
[5] Lacasa L, Luque B, Ballesteros F, Luque J and Nuno J C 2008 Proc. Natl Acad. Sci. USA105 4972 · Zbl 1205.05162 · doi:10.1073/pnas.0709247105
[6] Luque B, Lacasa L, Ballesteros F and Luque J 2009 Phys. Rev. E 80 046103 · doi:10.1103/PhysRevE.80.046103
[7] Donner R V, Zou Y, Donges J F, Marwan N and Kurths J 2010 New J. Phys.12 033025 · Zbl 1360.90045 · doi:10.1088/1367-2630/12/3/033025
[8] Donner R V, Zou Y, Donges J F, Marwan N and Kurths J 2010 Phys. Rev. E 81 015101 · doi:10.1103/PhysRevE.81.015101
[9] Small M, Zhang J and Xu X 2009 Lect. Notes Inst. Comput. Sci. Soc. Inf. Telecommun. Eng.5 2078 · doi:10.1007/978-3-642-02469-6_84
[10] Li C B, Yang H and Komatsuzaki T 2008 Proc. Natl Acad. Sci. USA105 536 · doi:10.1073/pnas.0707378105
[11] Marwan N, Donges J F, Zou Y, Donner R V and Kurths J 2009 Phys. Lett. A 373 4246 · Zbl 1234.05214 · doi:10.1016/j.physleta.2009.09.042
[12] Liu C and Zhou W X 2010 J. Phys. A: Math. Theor.43 495005 · doi:10.1088/1751-8113/43/49/495005
[13] Gao Z-K and Jin N-D 2009 Chaos19 033137 · doi:10.1063/1.3227736
[14] Xie W J and Zhou W X 2011 Physica A 390 3592 · doi:10.1016/j.physa.2011.04.020
[15] Ni X-H, Jiang Z-Q and Zhou W-X 2009 Phys. Lett. A 373 3822 · Zbl 1234.05216 · doi:10.1016/j.physleta.2009.08.041
[16] Qian M-C, Jiang Z-Q and Zhou W-X 2010 J. Phys. A: Math. Theor.43 335002 · doi:10.1088/1751-8113/43/33/335002
[17] Lacasa L, Luque B, Luque J and Nuño J C 2009 Europhys. Lett.86 30001 · doi:10.1209/0295-5075/86/30001
[18] Elsner J B, Jagger T H and Fogarty E A 2009 Geophys. Res. Lett.36 L16702 · doi:10.1029/2009GL039129
[19] Yang Y, Wang J-B, Yang H-J and Mang J-S 2009 Physica A 388 4431 · doi:10.1016/j.physa.2009.07.016
[20] Liu C, Zhou W-X and Yuan W-K 2010 Physica A 389 2675 · doi:10.1016/j.physa.2010.02.043
[21] Shao Z-G 2010 Appl. Phys. Lett.96 073703 · doi:10.1063/1.3308505
[22] Dong Z and Li X 2010 Appl. Phys. Lett.96 266101 · doi:10.1063/1.3458811
[23] Ahmadlou M, Adeli H and Adeli A 2010 J. Neural Transm.117 1099 · doi:10.1007/s00702-010-0450-3
[24] Tang Q, Liu J and Liu H-L 2010 Mod. Phys. Lett. B 24 1541 · Zbl 1196.37124 · doi:10.1142/S0217984910023335
[25] Yu Z G, Anh V, Eastes R and Wang D L 2012 Nonlinear Process. Geophys.19 657 · doi:10.5194/npg-19-657-2012
[26] Zhou Y W, Liu J L, Yu Z G, Zhao Z Q and Anh V 2014 Physica A 416 21 · Zbl 1395.92118 · doi:10.1016/j.physa.2014.08.047
[27] Mandelbrot B B 1983 The Fractal Geometry of Nature (New York: Academic)
[28] Makse H A, Davies G W, Havlin S, Ivanov P C, King P R and Stanley H E 1996 Phys. Rev. E 54 3129 · doi:10.1103/PhysRevE.54.3129
[29] Rybski D, Buldyrev S V, Havlin S, Liljeros F and Makse H A 2012 Sci. Rep.2 560 · doi:10.1038/srep00560
[30] Liu J L, Yu Z G and Anh V 2014 Phys. Rev. E 89 032814 · doi:10.1103/PhysRevE.89.032814
[31] Feder J 1988 Fractals (New York: Plenum) · Zbl 0648.28006 · doi:10.1007/978-1-4899-2124-6
[32] Falconer K 1997 Techniques in Fractal Geometry (New York: Wiley) · Zbl 0869.28003
[33] Canessa E 2000 J. Phys. A: Math. Gen.33 3637 · Zbl 0954.91045 · doi:10.1088/0305-4470/33/19/302
[34] Anh V V, Tieng Q M and Tse Y K 2000 Int. Trans. Oper. Res.7 349 · doi:10.1111/j.1475-3995.2000.tb00204.x
[35] Yu Z G, Anh V and Lau K S 2001 Phys. Rev. E 64 031903 · doi:10.1103/PhysRevE.64.031903
[36] Yu Z G, Anh V and Lau K S 2003 Phys. Rev. E 68 021913 · doi:10.1103/PhysRevE.68.021913
[37] Yu Z G, Anh V and Lau K S 2004 J. Theor. Biol.226 341 · Zbl 1439.92148 · doi:10.1016/j.jtbi.2003.09.009
[38] Yu Z G, Anh V, Lau K S and Zhou L Q 2006 Phys. Rev. E 73 031920 · doi:10.1103/PhysRevE.73.031920
[39] Yu Z G, Anh V V, Wanliss J A and Watson S M 2007 Chaos Solitons Fractals31 736 · doi:10.1016/j.chaos.2005.12.046
[40] Yu Z G, Anh V and Eastes R 2009 J. Geophys. Res.114 A05214 · doi:10.1029/2008JA013854
[41] Yu Z G, Anh V, Wang Y, Mao D and Wanliss J 2010 J. Geophys. Res.115 A10219 · doi:10.1029/2009JA015206
[42] Yu Z G, Anh V and Eastes R 2014 J. Geophys. Res.: Space Phys.119 7577 · doi:10.1002/2014JA019893
[43] Furuya S and Yakubo K 2011 Phys. Rev. E 84 036118 · doi:10.1103/PhysRevE.84.036118
[44] Song C, Gallos L K, Havlin S and Makse H A 2007 J. Stat. Mech. P03006 · doi:10.1088/1742-5468/2007/03/P03006
[45] Wang D L, Yu Z G and Anh V 2012 Chin. Phys. B 21 080504 · doi:10.1088/1674-1056/21/8/080504
[46] Li B G, Yu Z G and Zhou Y 2014 J. Stat. Mech. P02020 · doi:10.1088/1742-5468/2014/02/P02020
[47] Gallos L K, Song C, Havlin S and Makse H A 2007 Proc. Natl Acad. Sci. USA104 7746 · doi:10.1073/pnas.0700250104
[48] Liu J L, Yu Z G and Anh V 2015 Chaos25 023103 · Zbl 1345.28018 · doi:10.1063/1.4907557
[49] Tél T, Fülöp Á and Vicsek T 1989 Physica A 159 155 · doi:10.1016/0378-4371(89)90563-3
[50] Chung F R K 1997 Spectral Graph Theory(CBMS Regional Conf. Series in Mathematics vol 92) (Providence, RI: American Mathematical Society) · Zbl 0867.05046
[51] Lin Y and Yau S T 2010 Math. Res. Lett.17 345 · Zbl 1232.31003 · doi:10.4310/MRL.2010.v17.n2.a13
[52] Gutman I and Zhou B 2006 Linear Algebra Appl.414 29 · Zbl 1092.05045 · doi:10.1016/j.laa.2005.09.008
[53] Albert R, Jeong H and Barabasi A L 1999 Nature401 130 · doi:10.1038/43601
[54] Lacasa L and Toral R 2010 Phys. Rev. E 82 036120 · doi:10.1103/PhysRevE.82.036120
[55] Halsey T C, Jensen M H, Kadanoff L P, Procaccia I and Shraiman B I 1986 Phys. Rev. A 33 1141 · doi:10.1103/PhysRevA.33.1141
[56] Floyd R W 1962 Commun. ACM5 345 · doi:10.1145/367766.368168
[57] Gleich D F A 2008 graph library for Matlab based on the boost graph library http://dgleich.github.com/matlab-bgl
[58] Atay F M, Biyikoglu T and Jost J 2006 Physica D 224 35 · Zbl 1130.37344 · doi:10.1016/j.physd.2006.09.018
[59] Abry P and Sellan F 1996 Appl. Comput. Harmon. Anal.3 377 · Zbl 0862.60036 · doi:10.1006/acha.1996.0030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.