复杂网络中的传播扩散过程源头定位是网络科学中一类重要的反问题并且具有重要的现实意义，它受到了不同领域科学家的高度关注，与不同领域的重大研究方向存在着密切关联，包括社会学领域中的谣言传播、工程学领域中电网上发生的级联故障传播、医学领域中的流行病传播、经济学领域的金融危机扩散、计算机科学领域的互联网病毒传播、环境科学领域空气中污染物的扩散等。快速、准确地推断这些传播扩散过程的源头位置，能够帮助我们及时制定控制危害蔓延的解决方案，降低传播造成的风险，有效避免不必要的损失。由于传播扩散过程的动力学模型复杂多样，观测信息稀疏且随机性比较大，网络类型多样、拓扑结构异质性较强，源头节点数目未知等复杂因素的存在，精确定位传播扩散过程源头是一个公认的难度较大的研究方向。我们将针对复杂网络源头定位中的以下几点重要问题展开讨论，包括复杂网络中具有不同初始时刻的多源头定位，未知传播模型的源头定位，源头的可定位性及观测节点选择，异质传播模型源头定位和量子网络源头定位。
      针对扩散模型，为了解决多源头不同初始时刻的源头定位问题，我们提出了基于反向传播的极大似然多源头定位方法。利用反向传播求得每个节点的虚拟感染时刻。由中心极限定理可知，源头节点的初始时刻服从高斯分布。因此我们利用每个节点作为源头的极大似然值，定义了节点的中心性指标，然后通过循环删除观测节点的方法精确得到源头节点位置。该算法不仅能够精确定位具有不同初始时刻的源头节点，还能求得源头的初始时刻。在模型网络和实际网络上的模拟结果和在H1N1病毒源头定位上的成功应用证明了该算法不仅能利用稀疏的观测信息精确定位具有不同初始时刻的多个源头节点，而且算法具有较强的鲁棒性。
     我们讨论了适用于任意传播模型的源头定位算法。该方法基于消息的传播时间和传播路径长度成正相关的思想。利用节点感染时间向量和节点到源头最短路径向量之间的斯皮尔曼和皮尔逊相关系数作为节点的中心性指标，节点的中心性越大，其作为源头的可能性越大。
我们研究了算法的鲁棒性，网络规模对源头定位精度的影响，我们还讨论的源头初始时刻的选择问题，并提出了初始时刻求解方法，研究了除网络平均度外其他性质对源头定位精度的影响。讨论了算法的可定位性问题，并提出了针对斯皮尔曼和皮尔逊的可定位观测节点选择策略。最后探讨了皮尔逊算法在多源头定位中的应用情况。
     面对未知传播模型的情况，我们利用有限的观测信息推断出传播模型的传播延时均值和标准差。因此可以将任意传播模型转换成扩散模型，然后利用推断值提出了三个适用于任意传播模型的源头定位方法。我们研究了算法的鲁棒性和可定位性问题。定义了可定位性指标，并提出了基于贪心算法的观测节点选择策略。我们提出的观测节点选择策略能明显提高源头定位精度。
     考虑到复合种群，多层网络，时变网络等复杂情况，我们详细讨论了四个异质性传播模型，包括异质扩散模型，异质SI模型，时变网络扩散模型，随机游走模型。并提出了一个适用于上述任意异质传播模型的源头定位方法：逐阶邻居筛选算法。我们将算法与其他未知传播模型算法进行比较，发现该算法在异质性模型上的溯源精度优于其他算法。最后提出了一个与逐阶邻居筛选算法对应的选择观测节点的策略：全阶邻居覆盖策略。该策略选择的观测节点能够覆盖所有节点的任意阶邻居。模拟结果表明利用全阶邻居覆盖策略得到的观测节点既能提高观测节点利用率又能提高源头定位精度。
     我们首次讨论了基于量子游走的量子网络源头定位的问题。首先将基于密度矩阵的量子系统演化方程转换为线性系统的形式，然后利用观测到的有限节点的概率幅，借助压缩感知理论，重构量子游走者的初始状态，进而达到定位游走者源头位置的目的。我们分别讨论了连续量子游走与离散量子游走对应的量子网络的源头定位问题。最后我们讨论了量子网络的可观和可控性，定义了量子网络节点的观测范围，并提出了量子网络源头定位观测节点的选择策略。

   The source localization of the propagation and diffusion process in complex networks is a kind of important inverse problem of network science and has important practical significance. It has attracted much attention from scientists in different fields and is closely related to many major research areas including rumor spreading in sociology, cascading failure propagation of power grid in engineering, epidemic diseases in the field of medicine, financial crisis spreading in economics, spreading of Internet virus in computer science, diffusion of pollutants in the air of environmental science, etc. Inferring the source of these spreading processes quickly and accurately can help us to develop timely solutions to control the spread of hazards, to reduce the risks caused by transmission and to effectively avoid unnecessary losses. Due to the complexity and diversity of the dynamic models of the propagation and diffusion process, the sparse and random observer information, the diversity of network types, the heterogeneity of topological structure, the unknown number of source nodes and other complex factors, locating the source of the propagation and diffusion process is recognized as a difficult research direction. We will discuss the following important issues in the source localization of complex networks, including multi-source localization with different initial time, source localization of unknown propagation models, observer nodes selection, source localization of heterogeneous propagation models and source localization of quantum network.
    Aiming at the diffusion model, we propose a maximum likelihood multi-source localization method based on back propagation. Use the back propagation to find the virtual infection time of each node. According to the central limit theorem, the initial time of the source node obeys the Gaussian distribution. Therefore, we use the maximum likelihood value of each node to propose the centrality index of the node, and then obtain the source node position accurately by cyclically deleting the observation node. This algorithm can not only accurately locate the source nodes with different initial time, but also find the initial time of the source. The simulation results on the model network and the actual network prove the effectiveness of the algorithm.
    According to the property that the infection time of a node is positively related to the shortest path from the node to the source node, we propose a universal source location algorithm. By calculating the Spearman correlation coefficient and Pearson correlation coefficient between the node infection time vector and the shortest path vector from the node to the source node as the centrality index of the node. We discuss the traceability of the source location method and propose a corresponding observation node selection strategy based on the greedy algorithm. Finally, we generalize the algorithm to the multi-source localization problem.
    In the case of an unknown propagation model, we infer the mean and standard deviation of the propagation delay of the message in the network according to the infection time of the observation node, so any arbitrary propagation model can be transformed into a diffusion model. Then based on the obtained message propagation delay, three universal source location algorithms are proposed. We give unified traceability indicators for the three algorithms, and propose an observation node selection strategy based on the greedy algorithm.
    Considering complex situations such as complex populations, multilayer networks, and time-varying networks, we propose four heterogeneous propagation models, including heterogeneous diffusion models, heterogeneous SI models, time-varying network diffusion models, and random walk models. A universal source localization method suitable for arbitrary propagation models is proposed: a stepwise neighbor selection algorithm. In order to improve the utilization of observation nodes, we propose a strategy for selecting observation nodes. The observation nodes selected by this strategy can cover any order of neighbors of all nodes.
    We have studied the source location of quantum networks based on quantum walks. Therefore, we transform the system evolution equations, and then use the limited observation information, combined with the compressed sensing method, to reconstruct the initial state of the quantum walker, and then to locate the source of the message. We have discussed the source localization of quantum networks corresponding to continuous quantum walks and discrete quantum walks. Finally, we discussed the observability and controllability of the quantum network, and proposed a strategy of location observer nodes of the quantum network from the perspective of complex network control.