Evolution Mechanism And Some Dynamical Processes On Complex Network

The recent decade has witnessed the birth of a new movement of interest and research in the study of complex network in domestic and international area. The researchers in physics, biology, mathematics and computer all dedicate to the study of complex networks. It's from the point of systematic view to study real systems, such as the Internet, power grid networks and telephone call networks, etc. In this dissertation, we use statistical theory and optimization approaches to investigate a number of problems in complex network. The main work is to study the relationship between the network structure and its functions, especially the effects of the network structure on the dynamical processes on it, such as network robustness to random failures and network synchronizability. The main work is summarized as follows:1. Propose a scale-free network model with tunable clustering. The fundermental work to study complex network is to understand and model the network structure, which would lead to a better understanding of its dynamical process. In this paper, based on the BA model, we perform a more real network evolution process. Combined with the character of local community and the process of adding edges and nodes with probability p, we construct a new class of scale-free network model with tunable clustering. With the mean-field theory, we get the power-law degree distribution and the relationship between the degree and clustering. Analytical expression and numerical results indicate that the degree distribution follows power law and the clustering coefficient can be tuned with parameter p. The results of analytical expression and numerical results are in consistent.2. Entropy optimization of scale-free network's robustness to random failures. Cohen et al. used percolation theory to study scale-free network's robustness to random failures. The main character of scale-free network is its heterogeneity. We observed that the more heterogeneous the degree distribution is, the more robust the network is. Soheterogeneous degree can be measured by entropy of the degree distribution. By optimizing the entropy of the degree distribution, we get the relationship between the entropy value, the degree distribution exponent, the network scale and the minimal degree, which can provide a possible design for a robust scale-free network. Compared with the percolation value, we find that they are in positive correlation. So the degree distribution entropy can be an effective measure of the network robustness.3. Optimization of network robustness to random failures. Considered the effect of the network structure on the network performance and the network performance can be improved by changing the network structure. We study the problem of how to improve the network performance and what character the network should have when the network cost is given. By analysizing the relationship between the network robustness to random failures, the heterogeneity and the network efficiency, we find that the more robust the network is, the higher the network efficiency is. By optimizing the network efficiency with tabu search, the statistical properties of the most robust network to random failures have been obtained. The numerical results indicate that a network with less number of hub nodes and nodes connect in a disassortative pattern will be more robust to random failures. Then, with the optimization results, a nongrowing model has been constructed. Analysizing the characters of the model, the results indicate that the network robustness to random failures and the efficiency of the information exchange have been greatly improved, while the network synchronizability has been reduced.4. Optimizing synchronizabilities of networks. The network structure affects the dynamical process greatly. Up to now, the previous works on network synchronizability have been focusing on the issue of a particular statistical property's effect on the network synchronizability, such as the average path length, the clustering coefficient and the degree correlation. Inspired by the idea of the optimization approach, the network structural properties corresponding to the approximately best synchronizability have been investigated with tabu search in the meaning of topological and geographical networks, respectively.In the meaning of topological network, the network is an entity defined in an abstract space. The objective function is the eigenvalue ratio R of Laplacian matrix,while the constraint condition is to keep the degree distribution unchanged, which can be realized by intercrossing a pair of edges randomly. The numerical results indicate that even the degree distribution is given, the network synchronizability can still be improved. After optimization, compared with the initial network, the optimal network is the one with lower clustering, lower modularity and the degree correlation decreasing to negative reals, that is, the nodes connect in a disassortative pattern. Then, the subgraphs of the optimal network have also been investigated. The results show that the optimal network contains less number of loops of size 3 to 5, compared with the initial network.In the meaning of geographical network, the geographical distance between nodes has been considered, which can be realized by embedding a scale-free network into 2-dimensional mesh. After optimization, the connection pattern among nodes also changed. The nodes would prefer connecting with the ones further away from them in geographical space, which will naturally reduce the clustering coefficient. The maximal betweeness has also been investigated in topological and geographical meaning, respectively. The numerical results indicate that the maximal betweeness has negative correlation with the network synchronizability. That is, the larger the maximal betweeness, the worse the network synchronizability, which has been observed in both topological and geographical networks.