We explore the problem of inferring the graph Laplacian of a weighted, undirected network from snapshots of a single or multiple discrete-time consensus dynamics, subject to parameter uncertainty, taking place on the network. Specifically, we consider three problems in which we assume different levels of knowledge about the diffusion rates, observation times, and the input signal power of the dynamics. To solve these underdetermined problems, we propose a set of algorithms that leverage the spectral properties of the observed data and tools from convex optimization. Furthermore, we provide theoretical performance guarantees associated with these algorithms. We complement our theoretical work with numerical experiments, that demonstrate how our proposed methods outperform current state-of-the-art algorithms and showcase their effectiveness in recovering both synthetic and real-world networks.