Logistic regression is one of the most basic methods for modeling the probability of a binary outcome based on a collection of covariates. However, the classical formulation of logistic regression relies on the assumption of independent sampling. This is often violated when the results interact through the underlying network structure. This requires the development of models that can simultaneously handle both network peer effects (resulting from neighbor interactions) and high-dimensional covariate effects. In this paper, we incorporate such dependencies into high-dimensional logistic regression models by introducing quadratic interaction terms, such as the Ising model designed to capture pairwise interactions from the underlying network. develop a framework for The resulting model can also be viewed as an Ising model where the node-dependent external field linearly encodes the high-dimensional covariate. We propose a penalized maximum pseudo-likelihood method for estimating network peer and covariate effects. Besides handling the high dimensionality of the parameters, this conveniently circumvents the computational difficulties of the maximum likelihood approach. As a result, our method is computationally efficient, and the estimates achieve classical high-dimensional consistency under various standard regularity conditions. In particular, our results show that, even under network dependence, we can consistently estimate model parameters at the same speed as classical logistic regression when the true parameters are sparse and the underlying network is not too dense. I mean As a result of our general results, we derive the percentage of consistency for different natural network models. We also develop efficient algorithms for computing estimates and validate theoretical results in numerical experiments.
