论文总字数：38186字
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\fancyhead[l]{\kaishu{~~~东~南~大~学~本~科~毕~业~论~文}}
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\begin{center}{\kaishu \zihao{2}{单根节点有向无环网络的同步能力研究}}\end{center}
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\begin{center}{\kaishu\zihao{4} 摘\ \ \ \ 要}
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{\kaishu{\quad
丰富的有向无环网络存在于自然生物、工业和社会网络系统中，如鸟群、无人驾驶车队和人类社会。
有向无环网络的同步行为是诸多学科的热门话题。本文研究向单根节点有向无环图添加反向连边对网络同步
能力造成的影响，我们发现反向连边需要满足一定条件才能改变网络同步能力，有效的反向连边带来的影响仅与
反向连边所在的子网相关，确切地说，网络同步能力的影响与网络大小或起点位置都无关，仅与跨度和经过节点的入度有关。
本文通过$Laplacian$矩阵特征谱的推导结合数值模拟对上述结论进行了分析和证明。
\par
本文首先分析了反向连边对最简单的链状网络、网格状网络、树形网络三种具有单根节点的有向无环网络
同步能力的影响。在$\lambda_{2}$和$\lambda_{max}$层面，网络同步能力与反向连边跨度相关，并且随着跨度而减弱。
表征网络收敛速率的$\lambda_{2}$减小，谱半径$\lambda_{max}$则有相对更复杂的变化，这种变化因跨度的奇偶性而呈现相反的趋势。
对于链状网络和树形网络，反向连边跨度为1时，网络收敛速率下降为0.382黄金分割点。
\par
其次，本文结合具体例子和数值模拟将讨论推向更加一般的单根节点有向无环网络。并非所有的反向连边都能对网络同步能力产生影响，
通过寻找满足特定条件的“茎”，可以更精准地添加有效反向连边。反向连边跨度对谱半径的影响更多体现在跨度较小的情况下，
随着跨度增大，对谱半径的调控更多依赖于节点入度的大小。反向连边经过的茎的流节点与合并节点对特征谱$\lambda_{2}$和$\lambda_{max}$
都有相反的影响，流节点有助于减慢同步过程，而合并节点有助于加速同步过程。
}
\vskip 1cm \noindent{\kaishu 关键词: \ \ 复杂网络,\ 网络同步,\ 有向无环网络,\ $Laplacian$矩阵,\ 谱分析 }
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\begin{center}{\heiti \ Research on synchronizability of single-root directed acyclic networks}\end{center}
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\begin{center}{\rm\zihao{4} Abstract}
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\par
Abundant directed acyclic graphs（DAGs） exist in natural biological, industrial, and social networked systems like bird flocks, driverless teams and human social societies.
synchronizability of directed acyclic networks has been a hot topic in the research community. In this thesis, we explore the effect of adding a reverse edge to the single-root directed acyclic graph on synchronizability of the network.
Only the reverse edge that meets certain conditions can make a difference, and the difference is merely related to the subnetwork containing the reverse edge. Properly speaking, the effect is solely related to the range and the in-degree of the nodes, which is independent of the network size, topology or the location of the reverse edge。
In this thesis, we have the above conclusions analyzed and proved by the spectral analysis of $Laplacian$ combined with numerical simulation.
\par
In this thesis, we first give the results of three kinds of typical single-root directed acyclic networks, the simplest chain, grid and tree network.
In the sense of $\lambda_{2}$ and $\lambda_{max}$, the consensus performance of the network decreases with the range of the reverse edge while the spectral radius $\lambda_{max}$ appears to be more complex, which has something to do with the parity of the range.
The chain and tree network both have a convergence rate dropping to 0.382, point of the golden mean, when the range is 1.
\par
Next, we come to a more general conclusion through combining specific examples and numerical simulations. Not all reverse edges could change the synchronizability of the network. We may add a reverse edge more precisely on the “stem” that meets certain conditions.
The shorter range show clearer effect on the spectral radius. As the range grows, the evolution of the spectral radius depends more on the in-degree of nodes.
The surpassed streaming nodes and surpassed merging nodes of the stem show opposite effect on the $\lambda_{2}$ and $\lambda_{max}$. The streaming nodes help to slow down the synchronization process while merging nodes are just the opposite.
\vskip 0.8cm \noindent{\rm Key Words:\ \ Complex network, \ synchronization, \ directed acyclic graphs, \ $Laplacian$, \ spectral analysis }
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\chapter{绪论 }
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\section{研究背景及意义}
上个世纪八十年代以来，信息技术有了长足的发展，我们的社会终于进入了互联网时代，随之有一门交叉学科--复杂性科学逐渐走入研究者的视线，它为系统科学的发展注入了活力。
大自然中许多复杂系统在一定程度上都可以被建模成复杂网络进行分析，比如日常生活中的电力网络、交通网络、航空网络、调度网络等等，因此复杂网络的理论
和应用在物理学、生物学、生态学、气候学、工程学和数学领域中吸引了广泛的研究兴趣。
\par
随着复杂网络研究的蓬勃发展，与之相关的研究对象、现象、理论方法、结论等等内容越来越多，其中网络同步是复杂网络的基本行为之一。同步是自然界广泛
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