To address the unknown node ordering in graph-level variational decoders, nodes can be ordered using a heuristic approach. For example, depth-first search or breadth-first search can be used, ordering nodes starting from those with the highest degree:

$p_{\theta}(G | \mathbf{z}_G) \approx \prod_{(u,v)\in \mathcal{V}} \tilde{A}^{\pi}[u,v] A[u,v]+(1-\tilde{A}^{\pi}[u,v]) (1-A[u,v])$

Alternatively, a small heuristic set of orderings can be considered, averaging the likelihood over these orderings:

$p_{\theta}(G | \mathbf{z}_G) \approx \sum_{\pi_i\in\{\pi_1,\dots,\pi_n\}}\prod_{(u,v)\in \mathcal{V}} \tilde{A}^{\pi_i}[u,v] A[u,v]+(1-\tilde{A}^{\pi_i}[u,v]) (1-A[u,v])$

These heuristic methods perform effectively in practice while remaining computationally tractable.