Network models are widely used to represent relations between actors or nodes. Recent studies of the network literature and graph model revealed various characteristics of the actors and how they influenced the characteristics of neighboring actors. The first methodology is motivated by formulating a large network through the Exponential Random Graph Model and applying a Bayesian approach through the reference prior technique to control the sensitivity of the inference and to get the maximum information from the model. We consider a large Amazon product co-purchasing network (customers who bought this item also bought other products), and the purpose is to show how the blending of the Exponential Random Graph Model and Bayesian Computation efficiently handles the estimation procedure and calculates the probability of certain graph structures.The second methodology we discuss is an approach to a network problem where the network adjacency structure remains unobserved, and instead we have a nodal variable that inherits a hidden network structure. The key assumption in this method is that the nodes are assumed to have a specific position in an Euclidean social space. The main analysis is based on three big U.S. auto manufacturers and their suppliers, and recent research has explored the differences of the financial markets and an emphasis has been given to reveal the strategic interactions among companies and their industry rivals and suppliers, all of which have important implications for some fundamental questions in the financial economics. Economic shocks are transmitted through the customer supplier network and the whole industry could be affected by these shocks as they can move through the links of the actors in an industry. We developed an algorithm that captures the latent linkages between firms based on sales and cost data that influence various financial decision-making issues and financial strategies.Finally, we extend the problem of network estimation to Bayesian variable selection whereby an observed adjacency structure between different regions has been considered. The main idea is to select relevant variables region-wise. We investigate this problem using a Bayesian approach by introducing the Bayesian Group LASSO technique with a bi-level selection that not only selects the relevant variable groups but also selects the relevant variables within that group. We use spike and slab priors, along with the Conditional Autoregressive structure among the model coefficients, which validates the spatial interaction among the covariates. Median thresholding is used instead of posterior mean to have exact zeros for the variables that are not relevant. We finally implement the problem in the auto industry data and incorporate more variables to see whether the estimated adjacency structure helps us to indicate the relevant variables over different manufacturers and suppliers.
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