Use Adjacency Matrix To Research Controlability And Reachability Of Complex Networks

Since 2000, Kleinberg first proved the searchability of Complex Networks, many excellent search algorithms were then presented one by one. These classic algorithms successfully used in the transmission of the Complex Networks' information at that time. But with the application of Complex Networks gradual deepening, the traditional method can not be applied to more network and lack universal application. To apply to more network environment, the paper studies the search of Complex Networks based on the adjacency matrix's each power.In this paper, the work done:1) Propose the improved method for the highest degree search strategy, based on the adjacency matrix's each power. The highest degree search strategy can be seen as using the information of the adjacent matrix's second power. Some improved algorithms, based on the clustering coefficient and the highest degree search, are using the information of the adjacent matrix's second and third powers. However, in this paper, it considers more factors, such as the sum of the adjacent matrix's each power.2) Use the adjacency matrix method to totally describe the internal relations between nodes. The degree centrality, betweenness centrality and eigenvectors centrality are often used as the indicators to measure the importance of network nodes. In this paper, the sum of the adjacency matrix's each power is used as an new indicator, changing the strategy to select the next node in the highest degree search algorithm.3) Conduct a simulation experiment based on the analysis of the new method's principle. The improved algorithm and the highest degree search algorithm were simulated and compared in different networks, and then changed the network scale and the clustering coefficient to compare the changes between the existing algorithm and the improved one.In this paper, the new idea and method is:An new search method is proposed based on matrix's each power. By using the sum of the adjacency matrix's each power to measure the importance of nodes, an improved algorithm for the highest degree search is proposed, changing the factor that the highest degree search algorithm simply relies on the neighbor node degree distribution to select. The adjacency matrix's each power totally describes the distance among nodes needing to reach each other, and using these can more effectively distinguish central nodes and edge nodes of the network. As an new indicator to measure the importance of network nodes, giving priority to the nodes which have more paths, it can improve that the highest degree search method may drop into partial center in the search process.Through in different networks, there are several comparative simulation experiments between the improved algorithm of highest degree search and commonly used algorithms. The result shows that the improved algorithm can not only quickly reach network center, but also have the less average search steps and the wider network applicability.The work in this paper reflects the adjacency matrix effectiveness in the study of Complex Networks. The proposed algorithm which is used to the search of Complex Networks, has a certain reference value.