Researches On Some Limit Problems In Complex Networks

The advent of the computer age has aroused interest.in the study of complex networks.There are many complex networks in the real world,such as social networks,telephone networks,power grids,the Internet,the World Wide Web and scientists citation networks.The analysis of complex network properties plays an important role in more and more disciplines,such as physics,biology,medicine,computational science and statistics.With the in-depth study of real networks,network researchers find that many networks have some common characteristics,such as small-world phenomenon,scale-free property and high clustering coefficient.Erd(?)s-Rényi model is the most classic random graph model,but Erd(?)s-Rény model does not have scale-free property and high clustering coefficient.Therefore,people propose graphical models that are more in line with real-world networks.In this paper,we will study random intersecting graphs and inhomogeneous random graphs.We deeply study the largest connected components of random intersection graphs.In the subcritical regime,the weak law of large numbers for the largest tree component is given,thus the lower bound of the largest connected component is obtained.When the largest connected component is a tree component,we use random walk method to get the upper bound of the largest connected component;When the largest connected component is not a tree component,we obtain the upper bound of the largest connected component of the random intersection graph by studying the tree structure and the symmetry of the random intersection graph.Thus,we obtain the weak law of large numbers for the largest connected component.In the supercritical regime,we construct a related Markov chain to get the limit property of the largest connected component.By studying the properties of the Markov chain,we obtain the central limit theorem of the largest connected component.We also study the limiting properties of the number of subgraphs in two random graph models.Edges and triangles are two basic subgraphs.We study the limiting property of the number of triangles in a random intersection graph.When the expectation of the number of triangles is bounded,we use the Stein method to obtain the speed at which the number of triangles converges to the Poisson distribution;when the expectation of the number of triangles tends to infinity,we use the Stein-Tikhomirov method to obtain the bound of the Kolmogorov distance between the distribution of the number of triangles and the normal distribution.For inhomogeneous random graphs,we study the limiting property of the number of edges.Each vertex is assigned a weight,and these weights are independent and identically distributed random variables.When vertex weights have an finite mean,the number of edges in a inhomogeneous random graph is O(n).When vertex weights have an infinite mean,the number of edges is O(n log n)and the limiting distribution of the number of edges is also established.