Modeling Graphs with Attributes

It is well known that real-world networks from many domains exhibit a large number of structural similarities, such as the form of the distributions of node degrees, the spectrum of the adjacency matrix, and the degree of interconnectedness. Some of these network statistics, such as the clustering coefficient, capture local behavior, while others, like the diameter, capture more global structure. After it was shown that many well-studied classes of stochastic network models, like Erdös-Renyi random graphs, do not accurately capture most of these properties, many researchers have proposed generative models that attempt to capture various aspects of network structure. The stochastic Kronecker graph model has been promising because it appears to capture many different properties reasonably well, rather than capturing one at the expense of the others, and because efficient algorithms exist for fitting the model to massive datasets. However, due to its rigid construction, it is difficult to modify or to extend to take advantage of richer datasets, such as node or edge-level attributes that may be available. This paper introduces and investigates binomial attribute graphs, a new model inspired by the Kronecker graph construction. Surprisingly, despite discarding a large amount of structure imposed by the Kronecker graph construction, the new model performs at least as well, and is much more easily modified, as it follows standard probabilistic semantics. Binomial attribute graphs are too simplistic to serve as general-purpose network models, but they provide new insight into why the Kronecker graph construction may work and suggest directions for improvement.