二维有限能量约束下最优导航问题的理论分析

黎勇; 钭斐玲; 樊瑛; 狄增如

doi:10.7498/aps.61.228902

摘要

最近, Li 等研究了在Kleinberg导航模型中引入总能量=cN约束后的最优导航问题, 其中为网络中所有长程连边的长度之和, c为正常数, N为网络节点总数.他们通过在1维和2维导航模型中的模拟结果推测, 在有限能量约束下Kleinberg导航模型中按照幂律方式添加长程连边的最优幂指数应该是=d+1, 其中d为导航模型的维数.本文在平均场理论下, 建立了2维有限能量约束下的导航过程的动态微分方程, 通过对该方程进行数学分析以及数值求解, 从理论上证明了当网络规模足够大且总能量相对较小时, 2维有限能量约束下的最优导航幂指数确实为=3, 这一结果证实了Li等之前的推测.

关键词:

Abstract

Recently, a certain total energy constraint =cN was introduced into the Kleinberg's navigation model, where is the total length of the long-range connections, c is a positive constant and N is the network size. The simulation results obtained in the one and two-dimensional cases indicate that with total cost restricted the optimal power-law exponent for adding extra long-range links between any two nodes seems to be =d+1, where d is the dimension of the underlying lattice in this paper. Based on mean field theory, the navigation process on the 2-dimensional cost constrained navigation model can be described by dynamical equations. Based on our theoretical analysis and the numerical results of the dynamical equations, we prove that for large networks and comparatively small total energy, the optimal power-law exponent is =3 for the two-dimensional case. Our results can perfectly correspond to simulations reported previously.

Keywords:

作者及机构信息

1.
北京师范大学管理学院系统科学系, 北京师范大学复杂性研究中心, 北京 100875

基金项目: 中央高校基本科研业务费专项资金、国家自然科学基金(批准号: 60974084, 61174150)和NCET-09-0228资助的课题.

Authors and contacts

1.
Department of Systems Science, School of Management and Center for Complexity Research, Beijing Normal University, Beijing 100875, China

Funds: Project supported by the Fundamental Research Funds for the Central Universities and NSFC (Grants Nos. 60974084, 61174150) and NCET-09-0228.

参考文献

[1]	Li S B, Wu J J, Gao Z Y, Lin Y, Fu B B 2011 Acta Phys. Sin. 60 050701 (in Chinese) [李树彬, 吴建军, 高自友, 林勇, 傅白白 2011 物理学报 60 050701]
[2]	Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 1313 (in Chinese) [许丹, 李翔, 汪小帆 2007 物理学报 56 1313]
[3]	Watts D J, Strogatz S H 1998 Nature 393 6684
[4]	Barabasi A L, Albert R 1999 Science 286 509
[5]	Girvan M, Newman M E J 2004 Proc. Natl. Acad. Sci. 99 7821
[6]	Du H F, Li S Z, Marcus W F, Yue Z S, Yang X S, 2007 Acta Phys. Sin. 56 6886 (in Chinese) [杜海峰, 李树茁, Marcus W F, 悦中山, 杨绪松 2007 物理学报 56 6886]
[7]	Milgram S 1967 Psycholgy Today 2 60
[8]	Travers J, Milgram S 1969 Sociometry 32 425
[9]	Dodds P S, Muhamad R, Watts D J 2003 Science 301 827
[10]	Kleinberg J 2000 Nature 406 845
[11]	Kleinberg J 2000 Proceedings of the thirty-second annual ACM symposium on Theory of computing 163-170
[12]	Roberson M R, Ben-Avraham D 2006 Phys. Rev. E 74 17101
[13]	Martel C, Nguyen V 2004 Proceedings of the Symposium on Principles of Distributed Computing, ed. Kutten, S. (ACM Press, New York) 179-188
[14]	Carmi S, Carter S, Sun J, Ben-Avraham D 2009 Phys. Rev. Lett. 102 238702
[15]	Caretta Cartozo C, De Los Rios P 2009 Phys. Rev. Lett. 102 238702
[16]	Yang H, Nie Y C, Zeng A, Fan Y, Hu Y Q, Di Z R 2010 EPL 89 5800
[17]	Li G, Reis S D S, Moreira A A, Havlin S, Stanley H E, Andrade Jr. J S 2010 Phys. Rev. Lett. 104 018701
[18]	Bianconi G, Pin P, Marsilli M 2009 Proc. Natl. Acad. Sci. 106 11433
[19]	Li Y, Zhou D, Hu Y Q, Zhang J, Di Z R 2010 EPL 92 58002
[20]	Hu Y Q, Li Y, Di Z R, Fan Y 2010 arXiv: 1010.18

施引文献

[1]	Li S B, Wu J J, Gao Z Y, Lin Y, Fu B B 2011 Acta Phys. Sin. 60 050701 (in Chinese) [李树彬, 吴建军, 高自友, 林勇, 傅白白 2011 物理学报 60 050701]
[2]	Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 1313 (in Chinese) [许丹, 李翔, 汪小帆 2007 物理学报 56 1313]
[3]	Watts D J, Strogatz S H 1998 Nature 393 6684
[4]	Barabasi A L, Albert R 1999 Science 286 509
[5]	Girvan M, Newman M E J 2004 Proc. Natl. Acad. Sci. 99 7821
[6]	Du H F, Li S Z, Marcus W F, Yue Z S, Yang X S, 2007 Acta Phys. Sin. 56 6886 (in Chinese) [杜海峰, 李树茁, Marcus W F, 悦中山, 杨绪松 2007 物理学报 56 6886]
[7]	Milgram S 1967 Psycholgy Today 2 60
[8]	Travers J, Milgram S 1969 Sociometry 32 425
[9]	Dodds P S, Muhamad R, Watts D J 2003 Science 301 827
[10]	Kleinberg J 2000 Nature 406 845
[11]	Kleinberg J 2000 Proceedings of the thirty-second annual ACM symposium on Theory of computing 163-170
[12]	Roberson M R, Ben-Avraham D 2006 Phys. Rev. E 74 17101
[13]	Martel C, Nguyen V 2004 Proceedings of the Symposium on Principles of Distributed Computing, ed. Kutten, S. (ACM Press, New York) 179-188
[14]	Carmi S, Carter S, Sun J, Ben-Avraham D 2009 Phys. Rev. Lett. 102 238702
[15]	Caretta Cartozo C, De Los Rios P 2009 Phys. Rev. Lett. 102 238702
[16]	Yang H, Nie Y C, Zeng A, Fan Y, Hu Y Q, Di Z R 2010 EPL 89 5800
[17]	Li G, Reis S D S, Moreira A A, Havlin S, Stanley H E, Andrade Jr. J S 2010 Phys. Rev. Lett. 104 018701
[18]	Bianconi G, Pin P, Marsilli M 2009 Proc. Natl. Acad. Sci. 106 11433
[19]	Li Y, Zhou D, Hu Y Q, Zhang J, Di Z R 2010 EPL 92 58002
[20]	Hu Y Q, Li Y, Di Z R, Fan Y 2010 arXiv: 1010.18

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