由于各种具有复杂相互作用的系统结构可以抽象的用网络来表示，在近二十年来，网络科学快速发展并成功的应用于生物、经济、社会科学等多种学科领域，对于各领域复杂网络中普适性规律的挖掘，一直是网络科学研究的重点。
网络科学研究的核心内容主要包括网络的拓扑结构和网络上的动力学行为两个方面。由于网络的结构与系统功能的实现密切相关，因此对于网络结构的研究是复杂网络研究的基础和重点。事实上，关于小世界网络和无标度网络的研究，就是通过对复杂网络新的结构特征的揭示，推动了网络科学的发展。对于网络结构特征的刻画，相比于微观的顶点度、集聚系数以及宏观的度分布、距离等刻画方法，中观尺度度量能更好地统一网络在细节与全局上的丰富信息，一直是大家关注的重点。社团结构、模体、K-壳等都可以看作是对网络中观尺度结构的刻画。已有的网络 P 矩阵本质上也是一种对于网络结构的中观尺度刻画，且具有较强的可解释性和相对较优的时间复杂度等多种优势，但却并未得到系统和深入的研究。因此，本文的主要工作集中于挖掘网络 P 矩阵的特点，并重点发展基于 P 矩阵刻画的网络结构差异量化方法以及差异方向性刻画和可视化方法，进而开展其在网络骨架提取问题上的应用。论文的主要工作和创新点如下：
1、发展和丰富了基于网络 P 矩阵的介观网络结构分析方法，对节点相似性、节点聚类、网络演化和网络分类这四个经典的网络结构问题给出了相应的研究结果。具体而言，本文分别从基于 P 矩阵的节点 P 向量和整个网络的 P 向量两个方面出发，在节点层面，首先利用节点距离分布设计了一种量化网络节点间相似性的新方法，并从特征向量中心性和传播能力两个方面验证了该方法的有效性，进而提出了一种基于节点相似性的网络节点聚类算法并发现了复杂网络中的层次结构。在网络整体结构层面，提出了一种基于 P 向量的网络分类方法并与经典的特征向量分类方法进行了对比，分类准确率和计算效率两个指标都说明了基于 P 向量的分类方法具有更佳的效果。综合而言，各种实验结果展现了新提出的基于 P 矩阵的网络分析算法的有效性，也说明了基于 P 矩阵的网络结构刻画的可行性和有效性。
2、发展了加权网络 P 矩阵的构造方法，并进一步提出了量化加权网络结构差异的 WD 度量，解决了原有 D 度量的差异量化方法只适用于无权网络的局限性。首先，利用加权网络的平均最短路径和对应无权结构下的平均最短路径之比，重新标度网络各节点间的加权距离，建立了统计各级离散近邻集的节点数的基础，从而将适用于无权网络的 P 矩阵推广到广泛存在的加权网络上。进一步，给出了一种加权网络补图的定义以及一种加权网络 Alpha 中心性的计算方法，进而建立了综合的量化加权网络结构差异的 WD 度量。最后，进行了多种模拟实验和实证研究，主要包括对比全连通网络和具有各种边密度的非全连通网络在附加四种常见的权重分布前后的网络结构差异，对比通过不同训练次数得到的不同神经网络，以及多个真实网络。结果表明了新度量能有效捕捉权值对网络结构的影响，量化加权网络间的结构差异，并适用于不同规模、甚至不连通的加权网络之间的比较。
3、基于网络结构差异化的研究，提出了网络距离中的方向性问题，并发展了刻画和可视化网络结构差异方向性的方法。已有关于网络结构差异的研究，对于差异度量都还仅仅停留在只关注大小的层面，而忽略了结构差异的方向。我们在网络结构 P 矢量的刻画基础上，建立了描述结构差异方向性的指标和方法，并将方向性的讨论融入到一种对多个网络在二维平面的近似排布算法设计之中，实现了网络结构差异方向性的可视化。算法以随机网络为基准网，实现了各网络与随机网间的距离大小和方向在二位空间中的几何定位，给出了一种综合了网络更为细致的结构信息的差异化描述。此外，在得到各网络在平面上的排布结果的基础上，我们引入K-means聚类并提出了一种网络聚类方法。结合多个真实网络的位置分布和聚类结果，能够帮助分析不同网络间相似的结构模式，从而探讨真实网络中隐藏的结构信息。
4、在基于 P 矩阵的 WD 度量的基础上，将网络骨架提取问题转化为一个寻找最小结构差异子网络的优化问题，进而提出了一种基于网络结构差异的网络骨架提取算法(DFSA)。为证实新方法的有效性并观察其特点，我们在多种模拟网和真实网上展开对比实验，比较了DFSA算法与经典的全局阈值和差异过滤的骨架提取方法在直径、平均边显著性、节点比、渗流阈值等对比指标上的表现，以及固定密度子网络中保留节点的重要性等。实验结果表明， DFSA 算法能够更好地保持原始网络的性质，特别在距离相关的属性上和异质性网络上。同时，与已有方法相比，新方法不忽视网络中点强度或边权较弱，但可能在连通性上起着关键作用的节点或连边，使网络骨架更大程度地保证了原始网络的结构性质。

Since various systems with complex interactions can be abstractly represented as the network, in the past two decades, network science has developed rapidly and has been successfully applied in many fields such as biology, economy, and social sciences. Mining the general laws of different complex networks in various fields has always been the focus of network science.
The studies of universal laws in the network mainly include two aspects: network topology structure and dynamics. Because the functions of the network highly depend on its structure, the study about network structure becomes the basic and focused topic. In fact, the research on small-world networks and scale-free networks has promoted the development of network science by revealing the new structural characteristics of complex networks. For the characterization of network structure, compared with the microscopic approaches such as node degree and clustering coefficient and macroscopic approaches such as degree distribution and distance,
mesoscopic measurement can better unify the detailed and global rich information of the network and has been the focus of attention. As mesoscopic approaches of community structure, motif and K-shell, the existing P matrix of network is essentially a mesoscopic measurement of network structure, in addition to advantages of strong interpretability and relatively low time complexity, but it has not been systematically and thoroughly studied. Therefore, the main work of this paper focuses on mining the characteristics of network P matrix, and the development of P-matrix-based methods including quantization of the network structure dissimilarity as well as the characterization and visualization of directivity of network dissimilarity, and then carries out its application in the extraction of network skeleton. The main work and innovations of the paper are as follows:
1、This paper develops and enriches the analysis methods about mesoscopic network structure based on network P-matrix, and gives corresponding results on four typical network structure problems: node similarity, node clustering, network evolution and network classification. Specifically, we conduct from two aspects of nodes' P-vector and the P-vector of the entire network. Specifically, from the node level, we first design an approach of quantifying the similarity between network nodes by using the distribution of their nearest neighbors, and verify its effectiveness through eigenvector centrality and propagation ability. And then, a node-clustering algorithm based on the new similarity metric is proposed and the hierarchical structure is discovered in complex network. In addition, at the network level, a network-classification method integrating the network's P-vector and the difference between the nodes' P-vector is proposed and compared with the traditional feature-vector method. The two indicators of classification accuracy and time consumption show the new P-vector-based classification method has better effect. Various experimental results show the effectiveness of the new network analysis methods based on network P-matrix, and also illustrate the feasibility and effectiveness of the P-matrix on network structure characterization.
2、We construct and extend the P-matrix to weighted network, and further put forward the WD-measure to quantify the structural dissimilarity between weighted networks, which deals with the limitation that D-measure was only suitable for unweighted networks. Firstly, the weighted distance between each pair of nodes of the network is rescaled by the ratio between the average shortest path of the weighted network and that of its binary counterpart, to count the number of nodes in different order neighbor sets. As a result, the P-matrix only applicable to the unweighted is extended to the widely existing weighted network. In addition, we give a definition and a calculation to the complement graph and alpha centrality of weighted network respectively, so that the WD measurement is established to quantify the difference of two weighted networks. Finally, we conduct a variety of simulation experiments, mainly including comparison between complete graphs with four common weight distributions and not, and graphs with various density, comparison between neural networks, and comparison between several real world networks. Experiments on simulated networks and real networks show that the new metric can effectively capture the influence of weights on network structure, and quantitatively measure the dissimilarity of weighted networks. Moreover, it is suitable for the comparison between networks which have different sizes and are even disconnected.
3、Based on the research of network structure differentiation, the directivity problem in network distance is proposed, and the method of describing and visualizing the directivity of network structure differentiation is developed. Considering that the existing network distance only focuses on the size but ignores the direction, we set up the index and method to describe the directivity of structural differences, and integrate the discussion of direction into the design of an approximate layout algorithm for multiple networks on a two-dimensional plane. Constructing a random network as the reference network, the algorithm realizes the geometric positioning combining the distance size and direction between each network and random network in two-dimensional space, and gives a description that integrates more detailed structure information of the network. In addition, after obtaining the layout result, we bring in the K-means clustering algorithm and propose a  network-clustering method. Combining the locations and clustering results of multiple real networks can help analyze the similar structural patterns between them, so as to explore the hidden structural information in the real network.
4、Based on the WD metric, we transform the skeleton extraction into an optimization problem of finding subnetworks with the minimum WD, and then propose a network skeleton extraction algorithm (DFSA) based on network structure differences. In order to verify the effectiveness of the new method and observe its characteristics, we conduct a variety of experiments on synthetic and real-world networks. We compare the DFSA with the classical global thresholding and the Disparity methods in some indicators including diameter, average link salience, node ratio and percolation threshold. In addition, we also observe the importance of retained nodes in subnetwork with fixed density. The results show that our method can retain the properties better, especially on distance-dependent attributes and network with stronger heterogeneity. Moreover, the DFSA method does not belittle weak nodes and edges that might play a vital role in connectivity，which ensures the skeleton can preserve the properties of the original network to a greater extent.