The history of deep learning has shown that human-designed problem-specific networks can greatly improve the classification performance of general neural models. In most practical cases, however, choosing the optimal architecture for a given task remains a challenging problem. Recent architecture-search methods are able to automatically build neural models with strong performance but fail to fully appreciate the interaction between neural architecture and weights. This work investigates the problem of disentangling the role of the neural structure and its edge weights, by showing that well-trained architectures may not need any link-specific fine-tuning of the weights. We compare the performance of such weight-free networks (in our case these are binary networks with {0, 1}-valued weights) with random, weight-agnostic, pruned and standard fully connected networks. To find the optimal weight-agnostic network, we use a novel and computationally efficient method that translates the hard architecture-search problem into a feasible optimization problem.More specifically, we look at the optimal task-specific architectures as the optimal configuration of binary networks with {0, 1}-valued weights, which can be found through an approximate gradient descent strategy. Theoretical convergence guarantees of the proposed algorithm are obtained by bounding the error in the gradient approximation and its practical performance is evaluated on two real-world data sets. For measuring the structural similarities between different architectures, we use a novel spectral approach that allows us to underline the intrinsic differences between real-valued networks and weight-free architectures.