A Research On Degree Of Node In Complex Network

Complex network is a new kind of systems science. It regards systems with complex macrostructure as a network, analyzes them structural characters, such as topological indices, cause of formation, evolution and applications. Popularity of complex network thanks to excellent researches on two models: one is the small-world network model introduced by Watts and Strogaz, it reveals the importance of average path length and clustering coefficient, the other one is the scale-free network model introduced by Albert and Barabasi, it reveals the relationship between complex networks and degree distribution. These three spectacular concepts play a key role in recent study and development of complex network theory, they and some other properties influenced by them reveal the explicit ordered facts behind complex phenomena. Researches of complex network are always aim to discover these facts.This paper primarily focus on two problems: how to validate the scale-free phenomenon and estimate the exponent? how to compute the weight of nodes of directed weighted graphs? For the first one, we design a new accurate method named max-ranking to measure power-law. A sufficient and necessary condition of power-law estimation on occasion of huge integral samples is proved. then we compare the average-relative-error by different methods, and found that by the ARE of max-ranking is minimum. For the other one, we discuss the algorithm of PageRank, which is the first and the most influential instance utilizes the information of structure into large scale technical computation. We consider it as Markov chain, then ameliorate original model to a new one with unique stable distribution; we discuss the computability of it and convergence speed of its iteration; we also research the stability of its iteration, and gained three error upper bounds under different perturbations of settings.