We propose AFTNet, a novel network-constraint survival analysis method based on the Weibull accelerated failure time (AFT) model solved by a penalized likelihood approach for variable selection and estimation. When using the log-linear representation, the inference problem becomes a structured sparse regression problem for which we explicitly incorporate the correlation patterns among predictors using a double penalty that promotes both sparsity and grouping effect. Moreover, we establish the theoretical consistency for the AFTNet estimator and present an efficient iterative computational algorithm based on the proximal gradient descent method. Finally, we evaluate AFTNet performance both on synthetic and real data examples.