Complex Networks Theory And The Approximate Calculation Of Weight Distribution With Time Series

For a long time, many people regard the systems, natural or man-made, arroud us as the network, such as transportation networks, electricity, personal relationship and so on. We have studied the systems for one century with the graph theory-a branch of mathematics. We found that many real networks have some common topological properties, i.e. small world and scale-free. These properties are different from the regular and random network. As what the physicists say, complex network is in between regalar and random network. In this decade, many people said that the researches on network is no progress. But, on the contrary, domestic and international researches for network development rapidly. Needless to say, complex networks should be the best choise for the new generation of graduate students and young researches in related fields.In this paper, three parts are involved. The first part describes the research history of complex networks. The history can be traced back to the "Koningberg Seven Bridges problem" in the 18th century.1736 Euler solved the problem of villagers' puzzle. He opened the research of graph theory-one branch of mathematics. On this basis, the random graph theory established by Erdos and Renyi is recognized as the primary research to the networks in mathematics. The topological properties of complex networks include the average path lengthy, clustering coefficient, degree and degree distribution and robust, fragile. The evolution of the network model includes the random network, small world networks and scale-free networks.The second part describes one common phenomenon:synchronization. Synchronization play an important role in laser systems, superconducting materials and communication systems. But there are negative impact, such as the bridge collapse caused by resonance. To understand the network synchronization, we must first understand the concept, reseach history, characteristics of chaos and the definiton of dynamic networks. Then we discuss the complete synchronization, the relationship between the network topology and synchronization, and how to improve complex network synchronization.The third part is our research subject, this work propose one method for estimating the weight distribution of a network with Gauss edge weights using a time series collected from controlled measurements of the stable dynamics of the network. Comparing the weight distribution obtained using this method with that using another scheme demonstrates that the former is effective and accurate for approximate calculation of the weight distribution P(w) of coupled oscillator networks with a broad coupling strength region. This method enables the topological properties of underlying networks to be determined from only a time series of measurements because it depends on no prior information about the dynamical system. In addition, the method is rather robust against noise.