复杂网络中RSP模型的自适应动力学行为研究
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摘要
复杂网络中的结构和动力学二者的自适应的过程的研究是近两年学者关注的热点问题。可以说,网络结构和动力学行为的相互耦合问题,开创了复杂系统研究的新局面。本文从网络的结构和动力学行为的自适应入手,重点讨论生态系统中网络上的动力学行为。主要有以下三部分内容:
1对一类三物种循环捕食RSP模型的动力学机制进行了全面的研究。我们通过系统的蒙特卡洛方法(MC)模拟得出,在物种占据正方格子的格点中,系统会进入一个自组织稳定态。当考虑到捕食关系的入长程关联后,在淬火小世界网络中,系统会进入一个全局振荡态。我们又从系统的局部动力学进行了研究,发现无论是在格子还是在淬火网络中,系统局部都有振荡行为的出现。因此我们提出在格子中,各个局部的振荡步调是不一致导致整体出现一个自组织稳定态,而在淬火网络中,各个局部振荡的同步性使得整体出现一个振荡态。
2把网络结构和动力学行为自适应相结合,在研究生态网络捕食模型中引入自适应行为。我们给物种一定的逃生概率,这种行为会使系统的动力学发生变化,也会引起网络结构发生了变化。结果发现系统会从一个自组织稳定态向全局振荡态演化,然后又进入物种随机分布的稳定态。我们对这一现象产生的原因作了深入研究,并提出RSP系统产生振荡的两个必要条件:第一,从空间关系上,系统必须有一定数量的长程关联存在;第二,从空间分布上,系统必须有一定大小的由物种聚集形成的畴存在。当同时满足这两个条件时,振荡就出现了。
3对物种空间的移动性和空间分布及动力学演化做了深入讨论,给定物种一个自适应迁移概率。提出当物种以一种自适应方式迁移时,不仅保持了物种空间上的聚集形式存在,也保存了物种多样性的存在,同时分析了这种移动性对物种密度、空位密度的影响。
The investigation of the adaptive process between structure and dynamics of the complex network becomes a hot topic in recent years. It can be said that this question creates a new situation for the study of the complex systems. Starting from the adaptation between structure and dynamics of the complex network, the focused discussion in this paper is the dynamic behavior and its influence in a certain biological system. The main contents are as follows:
1. We study the mechanism of a spatial rock-scissors-paper model in a square lattice and a quenched small-world network. The system exhibits a global oscillation in the quenched small-world network, but the oscillation disappears in the square lattice. We find that there is a local oscillation in the square lattice the same as in the quenched small-world network. We can speculate that in the square lattice, superposition between the local oscillations in different patches leads to global stabilization, while in the quenched small-world network, long-range interactions can synchronize the local oscillations, and their coherence results in the global oscillation
2. Combining the dynamic behavior and the structure of network, we induce a probability of escape behavior to species in the RSP model. Both of the dynamics and structure change when inducing this adaptive behavior. The result shows that the system evolves from a self-organized stable state into an oscillation state, then into another stable state which the species locate randomly. For investigating the mechanism of the dynamic behavior, we propose that the appearance of the oscillation in system must satisfy two necessary conditions: Firstly, viewing from the spatial relationship, there must be a certain number of long-range interactions in system. Secondly, viewing from the spatial distribution, there must be a certain number of domain clustering by species in system.
3. We discuss the influence of the mobility of species to the spatial distribution and the dynamics in the RSP model. The results show that when giving a probability of adaptive movement to the species, the species domain is preserved, and the biodiversity of the system is maintained. Furthermore, we also analyze the effects of this adaptive mobility to the density of the species and the hollow sites.
1对一类三物种循环捕食RSP模型的动力学机制进行了全面的研究。我们通过系统的蒙特卡洛方法(MC)模拟得出,在物种占据正方格子的格点中,系统会进入一个自组织稳定态。当考虑到捕食关系的入长程关联后,在淬火小世界网络中,系统会进入一个全局振荡态。我们又从系统的局部动力学进行了研究,发现无论是在格子还是在淬火网络中,系统局部都有振荡行为的出现。因此我们提出在格子中,各个局部的振荡步调是不一致导致整体出现一个自组织稳定态,而在淬火网络中,各个局部振荡的同步性使得整体出现一个振荡态。
2把网络结构和动力学行为自适应相结合,在研究生态网络捕食模型中引入自适应行为。我们给物种一定的逃生概率,这种行为会使系统的动力学发生变化,也会引起网络结构发生了变化。结果发现系统会从一个自组织稳定态向全局振荡态演化,然后又进入物种随机分布的稳定态。我们对这一现象产生的原因作了深入研究,并提出RSP系统产生振荡的两个必要条件:第一,从空间关系上,系统必须有一定数量的长程关联存在;第二,从空间分布上,系统必须有一定大小的由物种聚集形成的畴存在。当同时满足这两个条件时,振荡就出现了。
3对物种空间的移动性和空间分布及动力学演化做了深入讨论,给定物种一个自适应迁移概率。提出当物种以一种自适应方式迁移时,不仅保持了物种空间上的聚集形式存在,也保存了物种多样性的存在,同时分析了这种移动性对物种密度、空位密度的影响。
The investigation of the adaptive process between structure and dynamics of the complex network becomes a hot topic in recent years. It can be said that this question creates a new situation for the study of the complex systems. Starting from the adaptation between structure and dynamics of the complex network, the focused discussion in this paper is the dynamic behavior and its influence in a certain biological system. The main contents are as follows:
1. We study the mechanism of a spatial rock-scissors-paper model in a square lattice and a quenched small-world network. The system exhibits a global oscillation in the quenched small-world network, but the oscillation disappears in the square lattice. We find that there is a local oscillation in the square lattice the same as in the quenched small-world network. We can speculate that in the square lattice, superposition between the local oscillations in different patches leads to global stabilization, while in the quenched small-world network, long-range interactions can synchronize the local oscillations, and their coherence results in the global oscillation
2. Combining the dynamic behavior and the structure of network, we induce a probability of escape behavior to species in the RSP model. Both of the dynamics and structure change when inducing this adaptive behavior. The result shows that the system evolves from a self-organized stable state into an oscillation state, then into another stable state which the species locate randomly. For investigating the mechanism of the dynamic behavior, we propose that the appearance of the oscillation in system must satisfy two necessary conditions: Firstly, viewing from the spatial relationship, there must be a certain number of long-range interactions in system. Secondly, viewing from the spatial distribution, there must be a certain number of domain clustering by species in system.
3. We discuss the influence of the mobility of species to the spatial distribution and the dynamics in the RSP model. The results show that when giving a probability of adaptive movement to the species, the species domain is preserved, and the biodiversity of the system is maintained. Furthermore, we also analyze the effects of this adaptive mobility to the density of the species and the hollow sites.
引文
[1] Albert R, Barabási A -L. Statistical mechanics of complex networks. Rev. Mod. Phys. 2002, 74: 47-97.
[2] Newman M E J. The structure and function of complex networks. SIAM Rev. 2003, 45: 167-256.
[3] Watts D J, Strogatz S H. Collective Dynamics of‘Small-World’Networks. Nature (London), 1998, 393: 440-442.
[4] Barabási A -L, Albert R. Emergence of scaling in random networks. Science, 1999, 286: 509-512.
[5] Dorogovtsev S N and Goltsev A V. Critical phenomena in complex networks. Rev. Mod. Phys. 2008, 80: 1275
[6]方锦清,汪小帆.一门崭新的交叉科学——网络科学.物理学进展: 2007, 03, 0239.
[7] Newman M E J,Watts D J. Scaling and percolation in the small-world network model. Phys. Rev. E 1999, 60:7332-7342.
[8] Dorogovtsev S N, Mendes J F F, Samukhin A N. Structure of growing networks with preferential linking. Phys. Rev. Lett. 2000.85:4633-4636.
[9] Holme P, Kim B J. Growing scale-free networks with tunable clustering. Phys. Rev. E 2003,67:056102.
[10] Barrat A,Barthélemy Vespignani M A. Weighted evolving networks: Coupling topology and weight dynamics. Phys. Rev. Lett. 2004,92:228701.
[11] Callaway D S, Newman M E J, Strogatz S H et al. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 2000, 85:5468-5471.
[12] Holme P, Kim B J. Vertex overload breakdown in evolving networks. Phys. Rev. E 2002, 65:066109.
[13] Barahona M, Pecora L M. Synchronization in small-world systems. Phys. Rev. Lett. 2002, 89:05410.
[14] Newman M E J.Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys. Rev. E 2001, 64, 016132.
[15] Newman M E J, Strogatz S H and Watts D J. Random graphs with arbitrary degree distributions and their applications, Phys. Rev E 2001, 64:026118.
[16] Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 2001, 86:3200-3203.
[17] Kuperman M, Abramson G. Small world effect in an epidemiological model. Phys. Rev. Lett. 2001, 86:2909-2912.
[18] Huang Li, Park K, Lai Y-C. Information propagation on modular networks. Phys. Rev. E 2006, 73:035103.
[19] Moreno Y, Nekovee M, Pacheco A F. Dynamics of rumor spreading in complex networks. Phys. Rev. E 2003,69:066130
[20] Thilo Gross, Carlos J Dommar D’Lima, Bernd Blasius. Epidemic Dynamics on an Adaptive Network. Phys. Rev. Lett. 2006, 96:208701.
[21] Leah B Shaw, Ira B Schwartz. Fluctuating epidemics on adaptive networks. Phys. Rev. E 2008, 77: 066101.
[22] Fefferman N H, Ng K L. How disease models in static networks can fail to approximate disease in dynamic networks. Phys. Rev. E 2007, 76: 031919.
[23] Petter Holme, Gourab Ghoshal. Dynamics of Networking Agents Competing for High Centrality and Low Degree. Phys. Rev. Lett. 2006, 098701.
[24] Daichi Kimura Yoshinori Hayakawa. Coevolutionary networks with homophily and heterophily. Phys. Rev. E 2008, 78: 016103.
[25] Burda Z A. Martin Krzywick O C. Adaptive networks of trading agents. Phys. Rev. E 2008, 78: 046106.
[26] Zhang Guo-Qing, Wang Di, Li Guo-Jie. Enhancing the transmission efficiency by edge deletion in scale-free networks. Phys. Rev. E 2005, 76: 017101.
[27] Morgenstern O, Von Neumann J. Theory of Games and Economic Behavior.Princeton,New Jersey,USA:Princeton University Press, 1947.
[28] Maynard Smith J.Evolution and the theory of games.American Scientist,1976,64(1):41—45.
[29] Axelrod R.Hamilton W D.The evolution of cooperation..Science,1981,211(4489):1390—1396.
[30] Hauert C,De Monte S,HoPoauer J,et a1.Volunteering as red queen mechanism for cooperation in public goods games.Science,2002,296(5570):1l29—1132.
[31] Sugden R.The Economics of Rights,Cooperation and Welfare,Basic Blackwell.UK:Oxford Press.1986.
[32] Nowak M A, May R M.Evolutionary games and spatial chaos.Nature,1992,359(6398):826—829.
[33] SzabóG, Fáth G. Evolutionary games on graphs. Phy. Rep. 2007, 446:97-216.
[34] Sinervo B, Lively C M. The rock-paper-scissors game and the evolution of alternative male strategies. Nature, 1996,380:240-243.
[35] Drossel B,Schwabl F. Self-organized critical forest-fire model. Phys. Rev. Lett. 1992, 69:1629-1732.
[36] Durrett R, Levin S. Spatial aspects of interspecific competition.Theor.Pop.Biol. 1998.53:30-43
[37] Tainaka K. Lattice model for the Lotka-Volterra system. J.Phys.Soc.Jpn.1988,57:2588-2590.
[38] SzabóG, Szolnoki A, Izsák R. Rock-Scissors-Paper Game on Regular Small-World. J. Phys. A, 2004, 37: 2599~2609.
[39] Lotka A J. Analytical Note on Certain Rhythmic Relations in Organic Systems. Proc. Natl. Acad. Sci. U.S.A. 1920, 6: 410~415.
[40] Volterra V Le?on sur la Théorie Mathematique de la Lutte pour la Vie. Paris, Gauthier-Villars, 1931.
[41] ChatéH, Manneville P. Collective Behaviors in Spatially Extended Systems with Local Interactions and Synchronous Updating. Prog. Theor. Phys, 1991, 87:1~60.
[42] Tainaka K. Stationary pattern of vortices or string in biological systems. Phys. Rev. Lett. 1989, 63:2688.
[43] Kuperman M and Abramson G. Small world effect in an epidemiological model. Phys. Rev. Lett. 2001, 86:2909.
[44] Szolnoki A and SzabóG. Phase transitions for rock-scissors-paper game on different network. Phys. Rev. E 2004 ,70: 037102
[45] Ying Chong-yang, Hua Da-yin, Wang Lie-yan. Phase transitions for a rock–scissors–paper model with long-range-directed interactions. J. Phys. A. Math. 2007,40: 4477–4482.
[46] Masuda N and Konno N. Network with dispersed degree save stable coexistence of species in cyclic competition. Phys. Rev. E 2006, 74:066102.
[47] Reichenbach T, Mobilia M and Frey E.Mobility promotes and jeopardizes biodiversity in rock-paper-scissors game. Nature.2007,448:1046.
[48] Reichenbach T, Frey E.Instability of spatial pattern and its ambiguous impact on species diversity. Phys. Rev. Lett. 2008,101:05810.
[2] Newman M E J. The structure and function of complex networks. SIAM Rev. 2003, 45: 167-256.
[3] Watts D J, Strogatz S H. Collective Dynamics of‘Small-World’Networks. Nature (London), 1998, 393: 440-442.
[4] Barabási A -L, Albert R. Emergence of scaling in random networks. Science, 1999, 286: 509-512.
[5] Dorogovtsev S N and Goltsev A V. Critical phenomena in complex networks. Rev. Mod. Phys. 2008, 80: 1275
[6]方锦清,汪小帆.一门崭新的交叉科学——网络科学.物理学进展: 2007, 03, 0239.
[7] Newman M E J,Watts D J. Scaling and percolation in the small-world network model. Phys. Rev. E 1999, 60:7332-7342.
[8] Dorogovtsev S N, Mendes J F F, Samukhin A N. Structure of growing networks with preferential linking. Phys. Rev. Lett. 2000.85:4633-4636.
[9] Holme P, Kim B J. Growing scale-free networks with tunable clustering. Phys. Rev. E 2003,67:056102.
[10] Barrat A,Barthélemy Vespignani M A. Weighted evolving networks: Coupling topology and weight dynamics. Phys. Rev. Lett. 2004,92:228701.
[11] Callaway D S, Newman M E J, Strogatz S H et al. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett. 2000, 85:5468-5471.
[12] Holme P, Kim B J. Vertex overload breakdown in evolving networks. Phys. Rev. E 2002, 65:066109.
[13] Barahona M, Pecora L M. Synchronization in small-world systems. Phys. Rev. Lett. 2002, 89:05410.
[14] Newman M E J.Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Phys. Rev. E 2001, 64, 016132.
[15] Newman M E J, Strogatz S H and Watts D J. Random graphs with arbitrary degree distributions and their applications, Phys. Rev E 2001, 64:026118.
[16] Pastor-Satorras R, Vespignani A. Epidemic spreading in scale-free networks. Phys. Rev. Lett. 2001, 86:3200-3203.
[17] Kuperman M, Abramson G. Small world effect in an epidemiological model. Phys. Rev. Lett. 2001, 86:2909-2912.
[18] Huang Li, Park K, Lai Y-C. Information propagation on modular networks. Phys. Rev. E 2006, 73:035103.
[19] Moreno Y, Nekovee M, Pacheco A F. Dynamics of rumor spreading in complex networks. Phys. Rev. E 2003,69:066130
[20] Thilo Gross, Carlos J Dommar D’Lima, Bernd Blasius. Epidemic Dynamics on an Adaptive Network. Phys. Rev. Lett. 2006, 96:208701.
[21] Leah B Shaw, Ira B Schwartz. Fluctuating epidemics on adaptive networks. Phys. Rev. E 2008, 77: 066101.
[22] Fefferman N H, Ng K L. How disease models in static networks can fail to approximate disease in dynamic networks. Phys. Rev. E 2007, 76: 031919.
[23] Petter Holme, Gourab Ghoshal. Dynamics of Networking Agents Competing for High Centrality and Low Degree. Phys. Rev. Lett. 2006, 098701.
[24] Daichi Kimura Yoshinori Hayakawa. Coevolutionary networks with homophily and heterophily. Phys. Rev. E 2008, 78: 016103.
[25] Burda Z A. Martin Krzywick O C. Adaptive networks of trading agents. Phys. Rev. E 2008, 78: 046106.
[26] Zhang Guo-Qing, Wang Di, Li Guo-Jie. Enhancing the transmission efficiency by edge deletion in scale-free networks. Phys. Rev. E 2005, 76: 017101.
[27] Morgenstern O, Von Neumann J. Theory of Games and Economic Behavior.Princeton,New Jersey,USA:Princeton University Press, 1947.
[28] Maynard Smith J.Evolution and the theory of games.American Scientist,1976,64(1):41—45.
[29] Axelrod R.Hamilton W D.The evolution of cooperation..Science,1981,211(4489):1390—1396.
[30] Hauert C,De Monte S,HoPoauer J,et a1.Volunteering as red queen mechanism for cooperation in public goods games.Science,2002,296(5570):1l29—1132.
[31] Sugden R.The Economics of Rights,Cooperation and Welfare,Basic Blackwell.UK:Oxford Press.1986.
[32] Nowak M A, May R M.Evolutionary games and spatial chaos.Nature,1992,359(6398):826—829.
[33] SzabóG, Fáth G. Evolutionary games on graphs. Phy. Rep. 2007, 446:97-216.
[34] Sinervo B, Lively C M. The rock-paper-scissors game and the evolution of alternative male strategies. Nature, 1996,380:240-243.
[35] Drossel B,Schwabl F. Self-organized critical forest-fire model. Phys. Rev. Lett. 1992, 69:1629-1732.
[36] Durrett R, Levin S. Spatial aspects of interspecific competition.Theor.Pop.Biol. 1998.53:30-43
[37] Tainaka K. Lattice model for the Lotka-Volterra system. J.Phys.Soc.Jpn.1988,57:2588-2590.
[38] SzabóG, Szolnoki A, Izsák R. Rock-Scissors-Paper Game on Regular Small-World. J. Phys. A, 2004, 37: 2599~2609.
[39] Lotka A J. Analytical Note on Certain Rhythmic Relations in Organic Systems. Proc. Natl. Acad. Sci. U.S.A. 1920, 6: 410~415.
[40] Volterra V Le?on sur la Théorie Mathematique de la Lutte pour la Vie. Paris, Gauthier-Villars, 1931.
[41] ChatéH, Manneville P. Collective Behaviors in Spatially Extended Systems with Local Interactions and Synchronous Updating. Prog. Theor. Phys, 1991, 87:1~60.
[42] Tainaka K. Stationary pattern of vortices or string in biological systems. Phys. Rev. Lett. 1989, 63:2688.
[43] Kuperman M and Abramson G. Small world effect in an epidemiological model. Phys. Rev. Lett. 2001, 86:2909.
[44] Szolnoki A and SzabóG. Phase transitions for rock-scissors-paper game on different network. Phys. Rev. E 2004 ,70: 037102
[45] Ying Chong-yang, Hua Da-yin, Wang Lie-yan. Phase transitions for a rock–scissors–paper model with long-range-directed interactions. J. Phys. A. Math. 2007,40: 4477–4482.
[46] Masuda N and Konno N. Network with dispersed degree save stable coexistence of species in cyclic competition. Phys. Rev. E 2006, 74:066102.
[47] Reichenbach T, Mobilia M and Frey E.Mobility promotes and jeopardizes biodiversity in rock-paper-scissors game. Nature.2007,448:1046.
[48] Reichenbach T, Frey E.Instability of spatial pattern and its ambiguous impact on species diversity. Phys. Rev. Lett. 2008,101:05810.