复杂网络中地震模型自组织临界行为的研究
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摘要
本文简要地介绍了自组织临界性的原理、复杂网络中若干具有代表性的网络机制模型以及作为自组织临界性研究的一个典型的模型-地震模型的基本概念和基本原理。综述了目前国际上主要研究的几种地震模型的构造、驱动机制、倒塌规则、雪崩动力学行为及其进展。在此基础上,我们提出了两个新的地震模型,利用数值模拟的方法,对这些模型的临界标度行为和雪崩动力学进行了详尽地研究,得出了具有创新意义的有价值的结论。我们的结果与国际相关文献最新研究结果的比较分析表明,我们的工作不仅有助于地震行为的研究和分析,而且有助于复杂网络中自组织临界行为的一般规律的深入讨论。
首先,我们研究了基于随机网络的单变量地震模型(OFC模型)中不均匀性对系统临界行为的影响。对模型的数值模拟研究表明:系统的临界行为对不均匀性非常敏感,不同的非均匀点数会导致系统不同的临界行为。更确切地说,其它参数(最近邻数和耗散参数)都相同,而非均匀点数不同的两个模型,不属于同一普适类。其次,我们构建了在随机网络中能量随机分配的单变量地震模型。数值模拟结果表明:只有在守恒情况下,系统具有自组织临界性,而且临界行为满足简单的有限尺寸标度律;在系统能量守恒时,不同的网络拓扑结构不影响系统的临界行为;而当系统中有耗散时,模型没有标度行为。
最后,我们构建了建立在随机网络中的双变量地震模型。对模型的数值模拟结果表明:在很大的耗散范围内,模型具有自组织临界性,地震尺寸几率分布满足有限尺寸标度律,网络拓扑结构对系统的临界行为没有影响。而且地震持续时间及在持续时间内的平均地震尺寸之间存在一个幂律关系,这为相应于能量不守恒的模型具有自组织临界行为提供了进一步的证据。
总之,我们的研究结果表明,不同的空间拓扑结构并不改变系统的临界行为,而不同的不均匀性(在二维地震模型中主要表现为边界条件的不同)、倒塌规则和驱动机制一定会影响系统的临界行为,改变模型的普适类。
We introduce the basic concept and the principle of self-organized criticality, several typical mechanism models on complex network, and its typical model-earthquake model. We summarize the structure, driving mechanism, toppling rules, avalanche dynamics and the newest investigation of some main earthquake models. On this base, we establish two new earthquake models with different driving mechanisms and spatial topology, and we use numerical simulation, study in detail the critical scaling behavior and avalanche dynamics of these models, we get some innovative and valuable conclusion. Furthermore, we compare and analyze the obtained results with the relative results on the international literature; this is not only helpful to the study on earthquake model, but also useful to the whole investigation on self-organized criticality in complex system.
First, we consider the influence of the inhomogeneity on the critical behavior in the one variable earthquake model on random network. Our numerical study shows that, the critical behavior of the system is very sensitive to the inhomogeneity, different inhomogeneities can result in different critical behavior. More exactly, the two models which have the same nearest neighbors and dissipative parameters, but have different inhomogeneities are not in the same universality class.
Second, we establish the one variable earthquake model on the random network which the energy redistributed randomly. Our numerical study shows that: the model displays self-organized criticality when the system is conservative, and the avalanche size probability distribution of the system obeys simply finite size scaling. Furthermore, when the system is conservative, different spatial topologies don’t alter the critical behavior of the system. Whereas, when the system is nonconservative, the model does not display scaling behavior.
At last, we establish the two-variable earthquake model on the random network. We numerically study the critical behavior of the model: we find that our model exhibits self-organized critical deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling. Different spatial topologies don’t alter the critical behavior of the system. In addition, a power law relation between the size and the duration of an avalanche exists, providing further evidence of criticality in the conconservative system.
Our results show that different spatial topologies don’t alter the critical behavior of the system, but different inhomogeneities(this corresponds to different boundary conditions in the OFC model on a lattice), toppling rules and driving mechanisms alter the behavior of the system significantly and change the universality class of models.
首先,我们研究了基于随机网络的单变量地震模型(OFC模型)中不均匀性对系统临界行为的影响。对模型的数值模拟研究表明:系统的临界行为对不均匀性非常敏感,不同的非均匀点数会导致系统不同的临界行为。更确切地说,其它参数(最近邻数和耗散参数)都相同,而非均匀点数不同的两个模型,不属于同一普适类。其次,我们构建了在随机网络中能量随机分配的单变量地震模型。数值模拟结果表明:只有在守恒情况下,系统具有自组织临界性,而且临界行为满足简单的有限尺寸标度律;在系统能量守恒时,不同的网络拓扑结构不影响系统的临界行为;而当系统中有耗散时,模型没有标度行为。
最后,我们构建了建立在随机网络中的双变量地震模型。对模型的数值模拟结果表明:在很大的耗散范围内,模型具有自组织临界性,地震尺寸几率分布满足有限尺寸标度律,网络拓扑结构对系统的临界行为没有影响。而且地震持续时间及在持续时间内的平均地震尺寸之间存在一个幂律关系,这为相应于能量不守恒的模型具有自组织临界行为提供了进一步的证据。
总之,我们的研究结果表明,不同的空间拓扑结构并不改变系统的临界行为,而不同的不均匀性(在二维地震模型中主要表现为边界条件的不同)、倒塌规则和驱动机制一定会影响系统的临界行为,改变模型的普适类。
We introduce the basic concept and the principle of self-organized criticality, several typical mechanism models on complex network, and its typical model-earthquake model. We summarize the structure, driving mechanism, toppling rules, avalanche dynamics and the newest investigation of some main earthquake models. On this base, we establish two new earthquake models with different driving mechanisms and spatial topology, and we use numerical simulation, study in detail the critical scaling behavior and avalanche dynamics of these models, we get some innovative and valuable conclusion. Furthermore, we compare and analyze the obtained results with the relative results on the international literature; this is not only helpful to the study on earthquake model, but also useful to the whole investigation on self-organized criticality in complex system.
First, we consider the influence of the inhomogeneity on the critical behavior in the one variable earthquake model on random network. Our numerical study shows that, the critical behavior of the system is very sensitive to the inhomogeneity, different inhomogeneities can result in different critical behavior. More exactly, the two models which have the same nearest neighbors and dissipative parameters, but have different inhomogeneities are not in the same universality class.
Second, we establish the one variable earthquake model on the random network which the energy redistributed randomly. Our numerical study shows that: the model displays self-organized criticality when the system is conservative, and the avalanche size probability distribution of the system obeys simply finite size scaling. Furthermore, when the system is conservative, different spatial topologies don’t alter the critical behavior of the system. Whereas, when the system is nonconservative, the model does not display scaling behavior.
At last, we establish the two-variable earthquake model on the random network. We numerically study the critical behavior of the model: we find that our model exhibits self-organized critical deep within the nonconservative regime. The probability distribution for avalanche size obeys finite size scaling. Different spatial topologies don’t alter the critical behavior of the system. In addition, a power law relation between the size and the duration of an avalanche exists, providing further evidence of criticality in the conconservative system.
Our results show that different spatial topologies don’t alter the critical behavior of the system, but different inhomogeneities(this corresponds to different boundary conditions in the OFC model on a lattice), toppling rules and driving mechanisms alter the behavior of the system significantly and change the universality class of models.
引文
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[2] 张济中 编著. 分形, 清华大学出版社, 1995
[3] 陆同兴 编著. 非线性物理概论, 中国科学技术大学出版社, 2002: 196-249
[4] 王东生, 曹磊编著. 混沌、分形及其应用, 中国科学技术大学出版社, 1995: 93-155
[5] 周凌云, 王瑞丽, 吴光敏等编著. 非线性物理理论及应用,科学出版社,2000: 192-259
[6] 席德勋 编著. 非线性物理概论, 南京大学出版社, 2000: 67-89
[7] 李炜. 演化中的标度行为和雪崩动力学, 博士学位论文, 华中师范大学, 2001
[8] P. Bak, C. Tang, and K. Wiesenfeld, Self-Organized Criticality: an Explanation of the 1/f Noise, Phys. Rev. Lett., 1987, 59: 381–384
[9] P. Bak, C. Tang, and K. Wiesenfeld, Self-Organized Criticality, Phys. Rev. A ,1988, 38: 364–374
[10] B. Drossel, S. Clar and F. Schwabl, Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model, Phys. Rev. Lett., 1993, 71: 3739–3742
[11] P. Bak and K. Sneppen, Punctuated Equilibrium and Criticality in a Simple Model of Evolution, Phys. Rev. Lett., 1993, 71: 4083–4086
[12] M. Paczuski, S. Maslov, and P. Bak, Avalanche Dynamics in Evolution, Growth, and Depinning Models, Phys. Rev. E , 1996, 53: 414–443
[13] Kan Chen and Per Bak, Self-Organized Criticality in a Crack-Propagation Model of Earthquakes, Phys. Rev. A, 1991, 43: 625–630
[14] Zeev Olami, Hans Jacob S. Feder, and Kim Christensen, Self-Organized Criticality in a Continuous, Nonconservative Cellular Automaton Modeling Earthquakes, Phys. Rev. Lett., 1992, 68: 1244–1247
[15] R. Pastor-Satorras and A. Vespignani, Critical Behavior and Conservation in Directed Sandpiles, Phys. Rev. E, 2000, 62: 6195–6205
[16] 孙霞, 吴自勤, 黄昀 编著. 分形原理及其应用, 中国科技大学出版社, 2003: 161-183
[17] R. Karmakar and S.S. Manna, Directed Fixed Energy Sandpile Model, Phys. Rev. E 2004, 69: 067107
[18] V.B. Priezzhev and K. Sneppen, Multiple Scaling in a One-Dimensional Sandpile, Phys. Rev. E, 1998, 58: 6959–6963
[19] S. Lübeck and K.D. Usadel, Numerical Determination of the Avalanche Exponents of the Bak-Tang-Wiesenfeld Model, Phys. Rev. E,1997, 55: 4095–4099
[20] P.A. Robinson, Scaling Properties of Self-Organized Criticality, Phys. Rev. E, 1994, 49: 3919–3926
[21] S.D. Edney, P.A. Robinson, and D. Chisholm, Scaling Exponents of Sandpile-Type Models of Self-Organized Criticality, Phys. Rev. E, 1998, 58: 5395–5402
[22] Zhang Duan-Ming, Pan Gui-Jun and Li Zhi-Hua et al., Critical Behavior in Non-Abelian Deterministic Directed Sandpile, Phys. Lett. A, 2005, 338 :163–168
[23] Zhang Duan-Ming, Pan Gui-Jun, and Li Zhi-Hua et al., Moment Analysis of Different Stochastic Directed Sandpile Model, Phys. Lett. A, 2005, 337: 285–291
[24] D.M.Zhang, G.J.Pan, Y.J.Lei, Mechanisms of Avalanche Dynamics in a Stochastic Four-State Sandpile Model, Chin. Phys. Lett., 2003, 20(12): 2122-2125
[25] L .P. Kadanoff, S.R. Nagel, and L. Wu et al., Scaling and Universality in Avalanches, Phys. Rev. A , 1989, 39: 6524–6537
[26] C. Tebaldi, M.De Menech, and A. Stella, Multifractal Scaling in the Bak-Tang-Wiesenfeld Sandpile and Edge Events, Phys. Rev. Lett., 1999, 83: 3952–3955
[27] M. Rossi, R. Pastor-Satorras, and A. Vespignani, Universality Class of Absorbing Phase Transitions with a Conserved Field, Phys. Rev. Lett., 2000, 85: 1803-1806
[28] D. Hughes and M. Paczuski, Large Scale Structures, Symmetry, and Universality in Sandpiles, Phy. Rev. Lett., 2002, 88: 054302
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