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  • Complex network is of vital importance in many natural systems for it can describe different kinds of complex systems which contain a large number of units with nodes and edges separately representing the component units and the interaction between nodes. In this research, a financing complex network has been established. The nodes stand for companies and the edge is determined by the strength of correlation between nodes, which means the similarity between companies. The correlation between two nodes is determined by the correlation between selected indicators. According to the method proposed by Prof. Gao etal. [33], here we illustrate how to use the strength of correlation between indicators to establish the edges and then construct the complex network. We use indicators of company’s governance structure and financing produce characteristic vector. For each pair of characteristic vectors, Ti and Tj, the correlation coefficient can be written as: Cij=∑k=1m[Ti(k)−⟨Ti⟩]•[Tj(k)−⟨Tj⟩]∑k=1m[Ti(k)−⟨Ti⟩]2•∑k=1m[Tj(k)−⟨Tj⟩]2(1) M is the dimension of the characteristic vector and ⟨Tj⟩=∑k=1mTj(k)/M, ⟨Ti⟩=∑k=1mTi(k)/M. The elements Cij are limited to the interval −1 ≤ Cij ≤ 1 in which Cij, 0, and -1, respectively represent the perfect correlations, no correlation and perfect anti-correlations. C is a symmetric matrix and Cij characterize the state of the connection between node i and j. Lastly, selecting an important threshold rc and then adjacency matrix A can be formed by translating the correlation matrix C. The principles of conversion are as follow: Aij={0,|cij|≤rc1,|cij|≥rc(2) It means if |cij| ≥ rc, there will be an edge connecting node i and j, otherwise, there will be no edge. The complex network consists of all the nodes and edges and the corresponding topological structure can be revealed by the adjacency matrix A. Newman (2004) proposed an idea of modularity Q. It is a quality function which is used to test whether the division of vertices into communities is meaningful and then community structure can be established. [34]. Supposing a network has n vertices. For a certain network divided into two parts, we set si = 1 if the vertex i is in part 1 and si = −1 if the vertex i is in part 2. And we annotate Aij as the number of edges between vertices i and j. Although Aij could be some large values in certain situations, it is 0 or 1 in most cases (there is always 0 or 1 edge between two vertices). But the expected number of edges between vertices i and j is, if edges are placed at random, where ki and kj are the degrees of the vertices and m=12∑iki is the total number of edges in the network. The modularity can be written as Q=14m∑ij(Aij−kikj2m)SiSj=14mSTBS(3) where s is the vector whose elements are the si. The leading factor of 1/4m issued for compatibility with the previous definition of modularity. Then we can have a new real symmetric matrix B with elements Bij=Aij−kikj2m(4) which is called the modularity matrix. We are focusing on the properties of this matrix. For now, we just get know of the elements of each the rows and columns are sum up to zero, therefore there is always an eigenvector (1,1, 1…) with eigenvalue zero. Given Eq (3), we can let s as a linear combination of the normalized eigenvectors ui of B, therefore s=∑i=1naiui with. After that we have Q=∑iaiuiTB∑jajuj=∑i=1n(uiT•s)2βi(5) where βi is the eigenvalue of B mapping at eigenvector ui. Furthermore, we take away the leading factor of 1/4m for making it brevity. The algorithm described above uses only of the signs of the elements of the leading eigenvector and the magnitudes convey information. Vertices related to elements of larger magnitude make large contributions to the modularity, and conversely for small ones. It shows that changing a vertex corresponding to a small magnitude element from group to another makes little difference to Q. Namely, elements’ magnitudes are a measurement of “how much” a vertex be a part of one community or another. That is to say, for vertices with elements near zero, they are on the borderline in communities. Therefore, the algorithm not only let us divides the vertices into groups, but let us changes them on a continuous scale of what degree they are part of one group or another. If a specific partition no longer produces within community edges and that would be expected by the random counterpart, Q = 0. Values other than 0 indicate deflection from randomness. The detection of community structure in complex network is of vital importance and attracts much attention. It may be the best thought of as a data analysis method used to illustrate the structure of network datasets such as social network in this paper. In this paper, the approach for detecting the community structure is based on the approach proposed by Newman (2006) [35]. Normally, community structure approaches assume network of interest that naturally divides into subgroups and the purpose is to find out these groups. Besides, community structure approaches may particularly recognize the possibility that there exists no good division of the network, a result that is itself considered to be of interest for its influence on the topology of the network. In consideration of the freedom that we can choose the size of our groups of vertices, the greatest value of the coefficient (u1T) in this term can be achieved. Dividing the vertices according to the signs of the elements in u1, following the principle that all vertices whose corresponding elements in u1 are positive go into one group and the rest in the other group. The algorthm is: we computing the major eigenvector of the modularity matrix and according to the signs of corresponding elements in this vector divide the vertices into groups. The connections of resulting network are characterized by the threshold rc. The pairs of nodes with weak correlations will be connected even if rc is extremely small. Under such situation, the physical content of correlations in time series will be hidden. The number of connections among nodes will reduce with the value of rc increasing and more noise will be revealed. However, heretofore, there has not been a general method used for determining the critical threshold. In this research, we can estimate a range of rc where the structure of complex network is relatively stable and we use Q to select the critical threshold rc. Especially, though simulating a specific process of the complex network, a critical threshold rc can be figured out. For example, decreasing the number of connections through increasing the value of rc and keeping the modularity which determine network almost unchanged. According to Newman’ algorithm, when values of Q are greater than 0.3, it reflects significant community structure, and then we can find out the threshold [34].
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