This dissertation mainly discussed topics related to power law graphs, including graph similarity testing algorithms and power law generative models. For graph similarity testing, we proposed a method based on the mathematical theory of diffusion over manifolds using random walks over graphs. We show that our method not only distinguishes between graphs with different degree distributions, but also graphs with the same degree distributions. We compare the undirected power law graphs generated by Barabasi-Albert model and directed power law graphs generated by Krapivsky's model to the random graphs generated by Erdos-Renyi model. We also compare power law graphs generated by four different generative models with the same degree distribution. The comparison results show that, our method performs better compared to the traditional features like eigenvalue spectrum and degree distributions. To study the generative mechanism of bivariate power law data in social networks, we use Poisson Counter Driven Stochastic Differential Equation (PCSDE) models as mathematical tool. We propose three types of bivariate PCSDE models. We study the tail dependence of the models and compare the models to real data in social networks. Type 1 model with Markov on-off modulation generates tail dependence coefficient (TDC) with values either zero or one; while the Type 2 model with coupled growth has the values between zero and one. The first two types of models do not fit the real data in distribution. Type 3 model keeps the shared Poisson counter in Type 1, but uses independent Brownian motion components instead of independent Poisson counters. We show that second Type 3 model with $0<\gamma<1$ has fractional TDC and synthetic data fits the real data in distribution. We study the applications of our proposed bivariate models. At first, the connection between Type 3 model to the existing network growing models is discussed. By connecting the two, our model explains why correlated bivariate power law in directed growing networks. The idea of exponential growth and random stopping can also be used to explain the existence of power law in many other natural or man-made phenomenons. We show that bivariate power law data also exists in natural images. We propose a new generative model for self-similar images based on our second Type 3 model.