【作者】 王燕；

【导师】 傅新楚； 许新建；

【作者基本信息】 上海大学 ， 应用数学， 2014， 博士

【摘要】 复杂网络的研究正从数学和物理学不断渗透到生物学、信息科学、工程技术科学、以及社会科学等不同学科中,具备多学科交叉和融合的特征。对复杂网络上系统的动态性质即网络上的动力学行为进行深入探索,也是网络时代复杂性科学研究中的热点方向。本文综合利用概率统计、随机过程及微分方程等知识,重点研究了一类由活性驱使的特定动态网络上的偏好随机游动的动力学行为及不同网络结构下长程作用对线性量子系统和非线性量子系统的影响。主要内容如下：1.由第二章分析知瞬时网络中各点的度与自身的活性成正比关系,我们选用通常的权重分配原则ωij＝(kikj)θ∝(aiaj)θ研究了此类动态网络上的偏好随机游动,即网络演化过程和随机游动过程同步进行。针对活点连边为无向边和有向边(出度边和入度边)三种情况,利用时齐Markov链、生成函数法等知识得到了平稳分布、任意两点间的平均首达时间和网络覆盖率的精确表达式,可以微调参数θ得到比无偏游动搜索策略更加有效得搜索全网的搜索策略。2.主要考虑了广义的游动原则对连续时间的量子随机游动的影响,即游动并不局限于最近邻邻居间的转移,而是任意两点间都可转移(无论有无连边),转移概率Γij是两点间最短距离的幂率衰竭函数,这种游走策略比一般的随机游走策略更能有效地遍历网络。基于这一思想,我们考虑到量子力学中长程作用随着粒子间的距离呈幂次逐次衰竭的原理,研究了在WS网络结构下广义游动原则在连续型量子游动中产生的不同影响：若不考虑断边概率p(=0),则远程作用并没有改变传播的有效性；而单纯增加断边概率p的值却会加剧出现局域化现象；若同时增加断边概率p及远程作用强度则进一步加剧1ocalization现象的出现。3.考虑了不同网络结构对非线性系统(也即非线性schrodinger方程)detrapping-trapping动力学变化的影响。分别针对含有参数∈的非线性schrodinger方程,主要利用数值方法考虑了规则环上次近邻作用。结果表明：∈=-1.0,N=5时耦合强度相同的次近邻作用的网络(最简单的全耦合网络),由长时间平均自返概率与非线性系数的函数关系知次近邻作用下初值问题稳定而最近邻作用下初值问题不稳定；∈=-1.0时,耦合强度相同的次近邻作用阻碍了方程self-trapping现象的发生；而当∈=1.0时,耦合强度相同的次近邻作用对方程的self-trapping现象几乎不影响；同时又考虑了规则环上断边重连引起的结构无序和远程作用两种因素不同程度地阻碍和促进了此两类非线性schrodinger方程的self-trapping现象的发生。更多还原

【Abstract】 The study of complex networks has permeated from Mathematics and Physics to Biology, Information science, Engineering, and Sociology, and so on, with inter-disciplinary and fusion features. The study of dynamics on complex networks has become an interesting topic in complexity. In this paper, with Probability Theory, Stochastic Process and Differential Equation, we studied the biased random walks on the activity-driven time-varying network, and investigated the dynamics of nonlinear and linear quantum systems on complex networks with long-range steps. The main results are as follows.1. Since the degree of any given node in the instantaneous network is proportional to its activity (see the results in chapter two), we assumed the weight ωij of edge linking node i and node j as ωij=(kikj)θ (αiαj)θ, such a form of edge weights can be observed in various real networks, and investigated the biased random walk on time varying networks. In other words, the dynamics of the network evolved and the random walk proceed at the same time scale. By using the knowledge of homogeneous Markov chain, generating function method, we obtained the exact expressions of the stationary distribution, the mean first passage time between any two nodes, and the coverage function for three different edge-construction cases. We can tune the values of parameter θ to get more effective search strategy than the unbiased walk search strategy.2. we introduced the generalized random-walk rule by considering that the transi-tion probability Γij, not restricted to nearest neighbors, allowing transitions that follow a power law as a function of the shortest distance(integer number of steps) between nodes, this navigation strategy is more efficient than the normal random walk strategy. As far as we know, the long-range interaction in quantum mechanics declined as a power law function of the interaction nodes’distance, thus we studied the continuous time quantum walks on WS network with the generalized random-walk rule compared with the random walk which only allowed the motion to the nearest neighbors. If only considering the long-ranged interaction, it gives no influence to the transport efficiency; only considering rewiring probability p, localization phenomenon obviously occurred; If simultaneously considering these two factors, the relatively stronger localization have become.3. In order to analyze how the different network structure affected the nonlinear system (i.e. nonlinear schrodinger equation), using numerical method, we mainly considered the self-trapping transition of nonlinear system on one-dimensional regular ring with next-neighbor interaction. For the case of∈=-1.0, N=5, the results showed that the system with next nearest neighbor interactions is unstable, however it is stable under the nearest neighbor interactions; For the case of∈=-1.0, the same coupling strength of next-neighbor interaction hindered the self-trapping transition, while it almostly has no effect for∈=1.0case.Moreover, we also investigated two factors, the rewiring probability p which induced disorder structure and the long-range interaction, how they affect the self-trapping transition of nonlinear systems respectively, compared to the case of considering these two factors simultaneously.更多还原