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We propose a method of estimating complex network topology with a noisy environment. Our method can estimate not only dynamical equation of the chaotic system and its parameters but also topology, the dynamical equation of each node, all the parameters, coupling direction and coupling strength of complex dynamical network composed of coupled unknown chaotic systems using only noisy time series. Estimating the system structure and parameter is regard as estimating the linear regression coefficients by reconstructing system with universal polynomial structure. Reconstruction algorithm of Bayesian compressive sensing is used for estimating the coefficients of regression polynomial. For the reconstruction from noisy time series we adopt relevance vector machine, namely we use sparse Bayesian learning to solve sparse undetermined linear equation to obtain the objects mentioned above. The Lorenz system and a scale free network composed of 200 Lorenz systems are provided to illustrate the efficiency. Simulation results show that our method improves the robust to noise compared with the compressive sensing and has fast convergence speed and tiny steady state error compared with the least square strategy.
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Keywords:
- sparse Bayesian learning /
- Bayesian compressive sensing /
- topology estimation /
- complex networks
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[2] Barabási A L, Albert R 1999 Science 286 509
[3] Fang X L, Jiang Z L 2007 Acta Phys. Sin. 56 7330 (in Chinese) [方小玲, 姜宗来 2007 物理学报 56 7330]
[4] Gao Z K, Jin N D 2009 Phys. Rev. E 79 066303
[5] Weng W G, Ni S J, Shen S F, Yuan H Y 2007 Acta Phys. Sin. 56 1938 (in Chinese) [翁文国, 倪顺江, 申世飞, 袁宏永 2007 物理学报 56 1938]
[6] He M H, Zhang D M, Wang H Y, Li X G, Fang P J 2010 Acta Phys. Sin. 59 5175 (in Chinese) [何敏华, 张端明, 王海艳, 李小刚, 方频捷 2010 物理学报 59 5175]
[7] Gao Z K, Jin N D, Wang W X, Lai Y C 2010 Phys. Rev. E 82 016210
[8] Liu M X, Ruan J 2009 Chin. Phys. B 18 2115
[9] Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 2109
[10] Jeong H, Tombor B, Albert R, Oltvai Z N, Barabasi A L 2000 Nature 407 651
[11] Guan X P, Peng H P, Li L X, Wang Y Q 2001 Acta Phys. Sin. 50 26 (in Chinese) [关新平, 彭海朋, 李丽香, 王益群 2001 物理学报 50 26]
[12] Wang X Y, Wu X J 2006 Acta Phys. Sin. 55 605 (in Chinese) [王兴元, 武相军 2006 物理学报 55 605]
[13] Li J F, Li N, Cai L, Zhang B 2008 Acta Phys. Sin. 57 7500 (in Chinese) [李建芬, 李农, 蔡理, 张斌 2008 物理学报 57 7500]
[14] Wang X Y, Meng J 2009 Acta Phys. Sin. 58 3780 (in Chinese) [王兴元, 孟娟 2009 物理学报 58 3780]
[15] Huang D 2004 Phys. Rev. E 69 067201
[16] Chen S H, Lu J H 2002 Phys. Lett. A 299 353
[17] Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴栋, 马西奎, 李富才, 尤勇 2002 物理学报 51 2459]
[18] Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高飞, 童恒庆 2006 物理学报 55 577]
[19] Li L X, Peng H P, Yang X Y, Wang X D 2007 Acta Phys. Sin. 56 51 (in Chinese) [李丽香, 彭海朋, 杨义先, 王向东 2007 物理学报 56 51]
[20] Alsing P M, Gavrielides A, Kovanis V 1994 Phys. Rev. E 49 1225
[21] Kobravi H R, Erfanian A 2009 Chaos 19 033111
[22] Zhou J, Lu J 2007 Physica A 386 481
[23] Wu X Q 2008 Physica A 387 997
[24] Chen L, Lu J A, Tse C K 2009 IEEE Trans. Circuits Syst.-II: Express Briefs 56 310
[25] Tang S X, Chen L, He Y G 2011 Chin. Phys. B 20 110502
[26] Yu D, Righero M, Kocarev L 2006 Phys. Rev. Lett. 97 188701
[27] Liu H, Lu J A, Lu J H, Hill D J 2009 Automatica 45 1799
[28] Gouesbet G, Letellier C 1994 Phys. Rev. E 49 4955
[29] Lu J A, Lu J H, Xie J, Chen G R 2003 Comput. Math. Appl. 46 1427
[30] Bezruchko B P, Smirnov D A 2000 Phys. Rev. E 63 016207
[31] Wang W X, Yang R, Lai Y C 2011 Phys. Rev. Lett. 106 154101
[32] Wang W X, Yang R, Lai Y C, Kovanis V, Harrison M A F 2011 EPL 94 48006
[33] Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[34] Candés E 2006 Proceedings of International Congress of Mathematicians Madrid, Spain, August 22-30 2006 p1433
[35] Candés E, Romberg J, Tao T 2006 IEEE Trans. Inform. Theory 52 489
[36] Duarte M F, Davenport M A, Takhar D, Sun T, Kelly K F, Baraniuk R G 2008 IEEE Signal Process. Magazine 25 83
[37] Bajwa W, Haupt J, Sayeed A, Nowak R 2007 IEEE Trans. Inform. Theory 53 3629
[38] Berger C R, Wang Z H, Huang J Z, Zhou S L 2010 IEEE Commun. Magazine 48 164
[39] Ji S H, Xue Y, Carin L 2008 IEEE Trans. Signal Process. 56 2346
[40] Tipping M E 2001 J. Mach. Learn. Res. 1 211
[41] Candés E, Tao T 2005 IEEE Trans. Inform. Theory 51 4203
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[1] Watts D J, Strogatz S H 1998 Nature 393 440
[2] Barabási A L, Albert R 1999 Science 286 509
[3] Fang X L, Jiang Z L 2007 Acta Phys. Sin. 56 7330 (in Chinese) [方小玲, 姜宗来 2007 物理学报 56 7330]
[4] Gao Z K, Jin N D 2009 Phys. Rev. E 79 066303
[5] Weng W G, Ni S J, Shen S F, Yuan H Y 2007 Acta Phys. Sin. 56 1938 (in Chinese) [翁文国, 倪顺江, 申世飞, 袁宏永 2007 物理学报 56 1938]
[6] He M H, Zhang D M, Wang H Y, Li X G, Fang P J 2010 Acta Phys. Sin. 59 5175 (in Chinese) [何敏华, 张端明, 王海艳, 李小刚, 方频捷 2010 物理学报 59 5175]
[7] Gao Z K, Jin N D, Wang W X, Lai Y C 2010 Phys. Rev. E 82 016210
[8] Liu M X, Ruan J 2009 Chin. Phys. B 18 2115
[9] Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 2109
[10] Jeong H, Tombor B, Albert R, Oltvai Z N, Barabasi A L 2000 Nature 407 651
[11] Guan X P, Peng H P, Li L X, Wang Y Q 2001 Acta Phys. Sin. 50 26 (in Chinese) [关新平, 彭海朋, 李丽香, 王益群 2001 物理学报 50 26]
[12] Wang X Y, Wu X J 2006 Acta Phys. Sin. 55 605 (in Chinese) [王兴元, 武相军 2006 物理学报 55 605]
[13] Li J F, Li N, Cai L, Zhang B 2008 Acta Phys. Sin. 57 7500 (in Chinese) [李建芬, 李农, 蔡理, 张斌 2008 物理学报 57 7500]
[14] Wang X Y, Meng J 2009 Acta Phys. Sin. 58 3780 (in Chinese) [王兴元, 孟娟 2009 物理学报 58 3780]
[15] Huang D 2004 Phys. Rev. E 69 067201
[16] Chen S H, Lu J H 2002 Phys. Lett. A 299 353
[17] Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴栋, 马西奎, 李富才, 尤勇 2002 物理学报 51 2459]
[18] Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高飞, 童恒庆 2006 物理学报 55 577]
[19] Li L X, Peng H P, Yang X Y, Wang X D 2007 Acta Phys. Sin. 56 51 (in Chinese) [李丽香, 彭海朋, 杨义先, 王向东 2007 物理学报 56 51]
[20] Alsing P M, Gavrielides A, Kovanis V 1994 Phys. Rev. E 49 1225
[21] Kobravi H R, Erfanian A 2009 Chaos 19 033111
[22] Zhou J, Lu J 2007 Physica A 386 481
[23] Wu X Q 2008 Physica A 387 997
[24] Chen L, Lu J A, Tse C K 2009 IEEE Trans. Circuits Syst.-II: Express Briefs 56 310
[25] Tang S X, Chen L, He Y G 2011 Chin. Phys. B 20 110502
[26] Yu D, Righero M, Kocarev L 2006 Phys. Rev. Lett. 97 188701
[27] Liu H, Lu J A, Lu J H, Hill D J 2009 Automatica 45 1799
[28] Gouesbet G, Letellier C 1994 Phys. Rev. E 49 4955
[29] Lu J A, Lu J H, Xie J, Chen G R 2003 Comput. Math. Appl. 46 1427
[30] Bezruchko B P, Smirnov D A 2000 Phys. Rev. E 63 016207
[31] Wang W X, Yang R, Lai Y C 2011 Phys. Rev. Lett. 106 154101
[32] Wang W X, Yang R, Lai Y C, Kovanis V, Harrison M A F 2011 EPL 94 48006
[33] Donoho D L 2006 IEEE Trans. Inform. Theory 52 1289
[34] Candés E 2006 Proceedings of International Congress of Mathematicians Madrid, Spain, August 22-30 2006 p1433
[35] Candés E, Romberg J, Tao T 2006 IEEE Trans. Inform. Theory 52 489
[36] Duarte M F, Davenport M A, Takhar D, Sun T, Kelly K F, Baraniuk R G 2008 IEEE Signal Process. Magazine 25 83
[37] Bajwa W, Haupt J, Sayeed A, Nowak R 2007 IEEE Trans. Inform. Theory 53 3629
[38] Berger C R, Wang Z H, Huang J Z, Zhou S L 2010 IEEE Commun. Magazine 48 164
[39] Ji S H, Xue Y, Carin L 2008 IEEE Trans. Signal Process. 56 2346
[40] Tipping M E 2001 J. Mach. Learn. Res. 1 211
[41] Candés E, Tao T 2005 IEEE Trans. Inform. Theory 51 4203
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