Bivariate Gamma Model

Among undirected graph models, the β-model plays a fundamental role and is widely applied to analyze network data. It assumes the edge probability is linked with the sum of the strength parameters of the two vertices through a sigmoid function. Because of the univariate nature of the link function, this formulation, despite its popularity, can be too restrictive for practical applications, even with a straightforward extension of the link function. For example, it is possible that vertices with similar strength parameters are more likely to be connected, in which case the edge probability depends on the distance of the strength parameters. Such common cases are not included in the β-model or its immediate extensions. In this paper, we propose a bivariate gamma model that links the edge probability with the two strength parameters by a symmetric bivariate function. The proposed model is more flexible than the β-model and its existing variants. It is also applicable to mirror various undirected networks. We show some special but representative cases of the bivariate gamma model by considering sparsity, mixture and other modifications, which cannot be properly handled by the β-model. Asymptotic theory is established to justify the consistency and asymptotic normality of the moment estimators. Numerical studies present evidence in support of the theory and an example involving real data further illustrates the applications.