Releasing the representations of nodes in real-world graphs associated with people or human-related activities, such as social and economic networks, gives adversaries a potential way to infer the sensitive information of edges. For example, graph convolutional layers initially aggregate node representations with their neighbors before passing them through non-linear activation functions. Hence, the released node representations may potentially breach edge privacy of the node neighbors. Thus, in this work, we study whether representations can be inverted to recover the graph used to generate them. We study three types of outputs that are trained on the graph, i.e., representations output from graph convolutional networks (GCNs), representations output from graph attention networks (GATs), and representations output from our proposed simplicial neural networks (SNNs). Unlike the first two types of representations that only encode pairwise relationships, the third type of representation, i.e., SNN outputs, encodes higher-order interactions (e.g., homological features) between nodes. We propose two graph reconstruction attacks (GRAs), i.e., Type-1 and Type-2 attacks, to recover a graph’s adjacency matrix from the three types of outputs trained on the graph. Specifically, our GRAs utilize a graph-decoder to minimize the reconstruction loss for the generated adjacency matrix via back-propagation. Our conclusions are two folds. First, our Type-2 attack achieves the best performance among all current GRAs. Second, we find that GCN outputs obtain the least precision and AUC on five datasets, followed by the GAT outputs, followed by the SNN outputs. Therefore, the SNN outputs reveal the lowest privacy-preserving ability to defend the GRAs. We further propose an unbiased multi-bit rectifier, by which the server can communicate with the nodes to privately collect their representations to defend the GRAs from potential adversaries.