复杂网络上动力学系统部分同步现象的研究

发布时间：2018-06-25 22:49

本文选题：复杂网络 + 网络平均距离　；参考：《广西师范大学》2016年硕士论文

【摘要】：从1998年Watts和Strogatz提出了小世界网络模型以及在1999年Barabasi和Albert提出了无标度网络模型之后,复杂网络的研究已经有了十余年的历史,随着网络科学及信息技术的发展,复杂网络的研究受到了生物、数学、物理、计算机信息科学等研究领域的专家学者们的青睐,它已经成为一个引起众多关注的交叉学科。网络的概念简而言之就是用节点代表个体,如果不同的个体间具有某种相互作用就在相应的节点间连接一条边,这条边可以根据相互作用的强度来分配权重。采用这种抽象的描述来研究存在相互作用的多个体间的行为可以使得研究的复杂性大大降低,使得探讨个体问的相互作用对整体行为的影响这一复杂的问题成为可能。理解复杂网络的结构特征对网络动力学行为(比如交通、传播、博弈和同步等)的影响是研究复杂网络的主要目标之一,复杂网络上动力系统的同步是其中的一项重要研究内容,尤其是介于网络的局域结构(小尺度)与网络整体(大尺度)之间的网络中尺度研究,具有重要的科学研究意义和实际应用价值。在本论文中,我们首先研究了网络的平均距离对网络复杂度的影响,然后分别探讨了在随机网络、无标度网络和群落网络处于最大复杂度时网络的动力学状态的变化情况,最后提出了几个描述网络部分同步状态的新方法,具体内容如下：(1)网络的平均距离是重要的网络结构特征量,它对网络的部分同步状态有着重要的影响,由于复杂度是刻画部分同步状态的重要特征量,所以我们研究网络的平均距离对复杂度的影响具有重要意义。通过数值模拟分析,我们发现当横纵差异比较大时,随着平均距离的增加复杂度是单调下降的；而当横纵差异比较小时,随着平均距离的增加复杂度是先升高后降低,并且耦合强度大的曲线首先出现这种非单调性。(2)我们还考察了网络具有不同的结构特点时的部分同步状态,并分析了处于部分同步状态时节点的分布特点,结果发现：若定义复杂度最大时对应的耦合强度为最优耦合强度,对于随机网络来说,处于最优耦合强度时就网络整体来说会使得节点集中但集中的中心不明显,节点仍然在圆环的全部范围分布；对无标度网络来说,处于最优耦合强度时就网络整体来说会使得节点集中但分布半径比随机网络大而集中的中心很明显,节点仍然在圆环的全部范围分布；对于群落网络来说,若外部边数量不多群落结构比较明显,处于最优耦合强度时节点显示出向多个中心聚拢,但聚拢中心在整个圆环上均匀分布。(3)由于生活中常见的同步现象一般是处于部分同步状态,对部分同步状态进行定量描述是一个重要的研究课题,由于之前设计的复杂度定义方法比较复杂,在本文我们提出了几种定义复杂度的新方法。我们已经发现随着耦合强度的增加,节点的位置分布越来越窄,而位置密度的曲线高度却越来越高,那么我们知道如果对这两类分布的宽度进行相乘,随着耦合强度的增加两者的乘积会先变大后变小,那么这个乘积会在一定程度上体现出部分同步状态。基于此,我们对部分同步状态做了如下定义：定义一—计算所有节点位置的占位比并计算相应的位置分布密度的跨分布区间的范围,将两者相乘：定义二—计算所有节点位置的占位比并计算相应的位置分布密度的占位比,将两者相乘；定义三—计算位置分布与密度分布两者的方差,将两者相乘：定义四—计算位置密度分布的熵与位置密度分布的分布的熵,将两者相乘。数值模拟结果表明该方法能够在一定程度上体现出节点的部分同步状态,但与之前定义的复杂度曲线在极大值的位置上存在错位,因此该定义有待于进一步深入的分析。
[Abstract]:The small world network model was proposed by Watts and Strogatz in 1998. After the scale free network model was put forward by Barabasi and Albert in 1999, the research of complex network has been over ten years. With the development of network science and information technology, the research of complex network has been studied by biology, mathematics, physics, computer information science and so on. In the field of experts and scholars, it has become an interdisciplinary subject of great concern. In short, the concept of network is to use nodes to represent individuals. If different individuals have some interaction, they connect one edge between the corresponding nodes. This edge can be used to assign weights according to the intensity of interaction. This abstract description of the interaction between multiple individuals can make the complexity of the study greatly reduced, making it possible to explore the complex problem of the impact of the interaction of individual questions on the overall behavior. The influence of step and so on is one of the main objectives to study the complex network. The synchronization of the power system on the complex network is one of the important research contents, especially the mesoscale research between the local structure (small scale) and the whole network (large scale), which has important scientific research significance and practical application value. In this paper, we first study the influence of the average distance of the network on the network complexity, and then discuss the changes in the dynamic state of the network when the stochastic network, the scale-free network and the community network are at the maximum complexity. Finally, several new methods to describe the synchronization state of the network division are proposed. The specific contents are as follows: (1) The average distance of the network is an important characteristic of the network structure. It has an important influence on the partial synchronization state of the network. Because the complexity is an important characteristic of the partial synchronization state, it is important to study the influence of the average distance of the network on the complexity. When the average distance increases, the complexity is monotonously decreasing with the increase of the average distance, while the complexity of the transverse and longitudinal difference increases first and then decreases with the increase of the average distance, and the non monotonicity is first appeared in the curve with large coupling strength. (2) we also examine the partial synchronism of the network with different structural characteristics. The distribution characteristics of the nodes in the partially synchronous state are analyzed. The results are as follows: if the corresponding coupling strength is the optimal coupling strength when defining the maximum complexity, for random network, the center of the network will not be concentrated but the center is not obvious when the optimal coupling strength is in the optimal coupling strength, and the node is still in the full circle. For the scale-free network, in the case of the scale-free network, in the optimal coupling strength, the network will make the node centralized but the center of the distribution radius is larger and concentrated than the random network. The node is still in the full range of the circle. For the community network, if the number of outer edges is not much more obvious, it is in the community network. In the optimal coupling strength, the nodes appear to gather in multiple centers, but the gathering center is evenly distributed on the entire circle. (3) because the common synchronization in life is generally in partial synchronization, the quantitative description of the partial synchronization state is an important research topic, and the complexity definition method of the previous design is more complex. In this article, we have proposed several new methods to define the complexity. We have found that with the increase of the coupling strength, the location of the nodes is narrower and narrower, and the height of the position density is getting higher and higher. Then we know that if the width of the two types of distribution is multiplied, the product of the coupling strength will increase with the increase of the coupling strength. Based on this, we define 1 - calculate the occupying ratio of all node positions and calculate the range of the cross distribution interval of the corresponding position distribution density, and multiply the two: definition two - all The occupying ratio of the node position and calculating the occupying ratio of the corresponding position distribution density and multiplying the two, defines three - the variance of both the calculated position distribution and the density distribution, and multiplies the two: defines the entropy of the entropy and the distribution of the position density distribution of the calculated position density distribution and multiplies the two. The numerical simulation results show the method. It can reflect the partial synchronization state of the node to a certain extent, but there is a dislocation with the location of the complexity curve defined previously in the maximum value, so the definition needs further in-depth analysis.
【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O157.5

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