Computing assortative mixing by degree with the s-metric in networks using linear programming

Calculation of assortative mixing by degree in networks indicates whether nodes with similar degree are connected to each other. In networks with scale-free distribution high values of assortative mixing by degree can be an indication of a hub-like core in networks. Degree correlation has generally been used to measure assortative mixing of a network. But it has been shown that degree correlation cannot always distinguish properly between different networks with nodes that have the same degrees. The so-called -metric has been shown to be a better choice to calculate assortative mixing. The -metric is normalized with respect to the class of networks without self-loops, multiple edges, and multiple components, while degree correlation is always normalized with respect to unrestricted networks, where self-loops, multiple edges, and multiple components are allowed. The challenge in computing the normalized -metric is in obtaining the minimum and maximum value within a specific class of networks. We show that this can be solved by using linear programming. We use Lagrangian relaxation and the subgradient algorithm to obtain a solution to the -metric problem. Several examples are given to illustrate the principles and some simulations indicate that the solutions are generally accurate

Objective Bayesian Edge Screening and Structure Selection for Ising Networks

The Ising model is one of the most widely analyzed graphical models in network psychometrics. However, popular approaches to parameter estimation and structure selection for the Ising model cannot naturally express uncertainty about the estimated parameters or selected structures. To address this issue, this paper offers an objective Bayesian approach to parameter estimation and structure selection for the Ising model. Our methods build on a continuous spike-and-slab approach. We show that our methods consistently select the correct structure and provide a new objective method to set the spike-and-slab hyperparameters. To circumvent the exploration of the complete structure space, which is too large in practical situations, we propose a novel approach that first screens for promising edges and then only explore the space instantiated by these edges. We apply our proposed methods to estimate the network of depression and alcohol use disorder symptoms from symptom scores of over 26,000 subjects. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11336-022-09848-8