Modeling Graphs Using a Mixture of Kronecker Models

Generative models for graphs are increasingly becoming a popular tool for researchers to generate realistic approximations of graphs. While in the past, focus was on generating graphs which follow general laws, such as the power law for degree distribution, current models have the ability to learn from observed graphs and generate synthetic approximations. The primary emphasis of existing models has been to closely match different properties of a single observed graph. Such models, though stochastic, tend to generate samples which do not have significant variance in terms of the various graph properties. We argue that in many cases real graphs are sampled drawn from a graph population (e.g., networks sampled at various time points, social networks for individual schools, healthcare networks for different geographic regions, etc.). Such populations typically exhibit significant variance. However, existing models are not designed to model this variance, which could lead to issues such as overfitting. We propose a graph generative model that focuses on matching the properties of real graphs and the natural variance expected for the corresponding population. The proposed model adopts a mixture-model strategy to expand the expressiveness of Kronecker product based graph models (KPGM), while building upon the two strengths of KPGM, viz., ability to model several key properties of graphs and to scale to massive graph sizes using its elegant fractal growth based formulation. The proposed model, called x-Kronecker Product Graph Model, or xKPGM, allows scalable learning from observed graphs and generates samples that match the mean and variance of several salient graph properties. We experimentally demonstrate the capability of the proposed model to capture the inherent variability in real world graphs on a variety of publicly available graph data sets.