If I understand correctly, the concept is to, first, embed the given graphs using the graph isomorphism network (GIN) [Xu et al.] and then, for the GED, feed any two embeddings $\mathbf{x}_{\mathcal{G}_1}$ and $\mathbf{x}_{\mathcal{G}_2}$ of graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ to the euclidean norm, i.e. $\text{GED}(\mathcal{G}_1, \mathcal{G}_2) = \parallel \mathbf{x}_{\mathcal{G}_1} - \mathbf{x}_{\mathcal{G}_2} \parallel_2$. This architecture is then trained to predict the graph edit distance.
To my understanding, the metric properties of this approach rely entirely on the expressive power of the GIN. If the GIN generates the exact same embedding for two dissimilar graphs, then $\text{GED}(\mathcal{G}_1, \mathcal{G}_2) = 0$ even though $\mathcal{G}_1 \neq \mathcal{G}_2$ which violates the properties of a metric. As GINs expressive power is upper bounded by that of the Weisfeiler Lehman test (WL-test) (cf. Xu et al.), you can easily find such pairs of graphs. For more on this, you can read this nice blog post.
At first glance, this seems to be contradicting the claim of the authors. However, as far as I understand, the authors only claim that $F$ preserves metric properties. Since $F$ is the p-norm, i.e. the GIN is excluded from the definition of $F$, they are correct. The authors don't claim that the full architecture including the GIN preserves metric properties.
To answer your questions:
I wonder how can I prove that these properties are maintained in actual forecasts?
You can only disproof that, because GIN cannot distinguish graphs that the WL-test cannot distinguish.
is it enough for satisfying properties at the real prediction value to define loss function that is always satisfying properties which i want?
Sadly no, this is not enough - neural networks are only function approximators, there is usually no guarantee that an NN will always satisfy the properties promoted by the loss function.
The bottom line is that if you truly rely on absolutely correct measurements, then you should not use this approach. However, exact measurements take a lot of time to compute (as the authors say it's an NP-hard problem). Therefore, this method presents a tradeoff between accuracy and time complexity. If the accuracy suffices, then it can be orders of magnitude faster than explicitly computing GED.
