The Topology Analysis Of Random Maximal Planar Networks

The rapid development of information technology makes human society entered the network age. From power networks to the global transportation network, from the Internet to the world wide web, from the brains to the metabolic network, from DNA, RNA to the protein network, from the scientific collaboration network to the social networks,we can say, people are living in a world full of a variety of complex networks. The network in the human society improves the quality of daily life and the production efficiency. But at the same time, it takes a certain degree of negative impact to the society. So, it requires people to have sufficient understanding of the variety of complex networks. And the object of the complex network theory is the general character and method to identify the complex networks. This article is mainly on the analysis of a class of complex networks that structured by some rules. We call them the random maximal planar networks.A graph is called a planar graph, if and only if the graph can be drawn in a plane, and none of its edges cross the others. In a simple planar graph, if you add an edge between any two adjacent nodes in the graph, the graph turns to a non-planar graph, then we call the graph a maximal planar graph. The planar networks play a very important role in our daily life. For example,in the design of the circuit,the connection between the internal element could not cross each other. So, we request to design the circuit board as a planar one. This article explains the process to construct a maximal planar network by the method which is a combination of the3-order and4-order methods. And it explained the topological properties of scale-free, high clustering coefficien-t and small average path length of the network. Then, we apply a5-order method to construct maximal planar networks, and create a new network. We also did a qualitative analysis on the clustering coefficient of the constructed networks. We also simulated the degree distribution of the networks that con-tain10000,50000and100000nodes by programming method, then draw them in the double logarithmic coordinate systems. Then we fitted the distribution curve to investigate the variation tendency of the clustering coefficient and average path length about the order of the networks. At last, we combined the3-order and5-order methods to construct some new maximal planar network-s. By programming method, we simulated the curves of degree distribution, clustering coefficient and average path length of the networks, and fitted the curve of the degree distribution, investigated the variation tendency and rules of the clustering coefficient and average path length about the order of the networks. By the results of the fittings,whether the5-order method, nor the combination of the3-order and the5-order methods, the degree distributions are power-law distributions, the distribution P(k)∝(k+c)-γ. And we also have the clustering coefficient C>0.4, the average path length L(N)∝In N. So, we can see the small world and scale-free properties of the new constructed networks.These properties are very consistent with the real networks.In addition to this article, we can promote such networks by some new methods. For example,the combination of4-order and5-order methods, the combination of3-order,4-order and5-order methods. Through these methods, we can construct some more complex networks. Then, we can also analyze the properties of these networks. We can relax the maximal planar networks to1-planar graphs. A graph is called1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. Follow this definition, when we add a new point to the network, the structure of the network is more complex and changeful.