We consider a scenario where a steady network state (output graph signal) has been generated by an initial network state (input graph signal) locally diffused through the edges of the network according to an auto-regressive and/or moving average dynamics of order one. The input graph signal can represent, for example, an initial opinion profile and the output the consensus opinion formed after individuals exchange their views with their friends. For that scenario, we analyze how a set of input-output pairs (each corresponding to a different opinion cascade) can be used to infer the topology of the underlying graph. The problem is formulated as a least squares minimization augmented with topological constraints, which include sparsity on the graph edges. While the original network recovery problem is non-convex, suitable convex relaxations along with theoretical recovery guarantees are presented. We first look at the case where all input-output pairs have been generated with the same graph but the diffusion coefficients for each observation are different. We then discuss the case where the graphs are related but not exactly the same. Numerical tests showcase the effectiveness of the proposed algorithms in recovering different types of networks with synthetic signals.