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A Hyper-parameters
We fine-tune the hyper-parameters for our model FourIE using the development data of the ACE05-E+ dataset (our main and largest evaluation dataset). The selection criteria is based on the average F1 scores of the four IE tasks of consideration (EME, ETD, RE, and EAE). The following values are suggested by the fine-tuning:  $2e - 5$  for the learning rate of BertAdam for the optimizer; 10 for the batch size;  $N_{i} = 2$  and  $N_{t} = 3$  for the numbers of layers in the GCN models  $\mathsf{G}^{\text{inst}}$  and  $\mathsf{G}^{\text{pred}}$  respectively; 300 hidden units for all the layers of the feed forward networks GCN models, and type embeddings;  $\beta = 100$  for the constant in the approximation  $\hat{\mathbf{A}}^{pred}$ ;  $\tau = 0.1$  for the temperature parameter; and  $\lambda = 0.5$  for the trade-off parameter in the loss function. To achieve a consistency, we apply the same hyper-parameters from this fine-tuning for other datasets.
Analysis. To better understand the contribution of the knowledge from the type dependency graphs  $\mathbf{G}^{gold}$  and  $\mathbf{G}^{pred}$  for FourIE on EAE, we analyze the set of all the argument instances on the ACE05-E+ development set (called  $\mathcal{A}$ ) that FourIE can successfully predict the argument roles while "FourIE-GCN $^{type}$ " fails to do so. In particular, we find three major categories of the instances in  $\mathcal{A}$  that highlight the benefits of the type dependency graphs:
(i) One-edge constraints (accounting for  $28.9\%$  of  $\mathcal{A}$ ): The incorrect argument role predictions of "FourIE-GCN $^{type}$ " for these instances violate the constraint on the possible argument roles of event types. As FourIE does not have this issue, it suggests that FourIE can learn and enforce those constraints (i.e., from the event Argument edges of  $\mathbf{G}^{gold}$ ) from the training. For instance, in the sentence "... the United States upped its military presence, deploying more missile-firing warships to the Red Sea", both FourIE and "FourIE-GCN $^{type}$ " can recognize "deploying" as an event trigger of type Transport. However, regarding the entity mention "Red Sea", FourIE correctly assigns the Destination role for the Transport event while "FourIE-GCN $^{type}$ " incorrectly considers it as the role Place (an invalid role for the event type Transport).
(ii) Two-edge constraints (representing  $36.5\%$  of  $\mathcal{A}$ ): The predictions from "FourIE-GCN $^{type}$ " in this category involve argument roles that are never assigned to an entity mention of some entity type
in an event mention/trigger of some event type. FourIE can avoid this issue as it can recognize these constraints from the combinations of two neighboring edges (i.e., an event Argument and and entity Argument edge). For example, in the sentence "... the tanks and Bradley fighting vehicles ... backed by the Apache attack helicopters ... punched through the Republican Guard defenses ..., both FourIE and "FourIE-GCN type," can detect "helicopters" as an entity mention of type Vehicle which is an argument for the "Attack" event triggered by "punched through". However, "FourIE-GCN type" incorrectly predicts the argument role of "Attacker" for "helicopters" while FourIE can successfully return the Instrument in this case. In fact, we cannot find any Vehicle entity that plays the Attacker role in an Attack event in the training data, providing an useful information for FourIE to learn and fix the error.
(iii) Four-edge constraints (accounting for  $19.2\%$  of  $\mathcal{A}$ ): The failure of “FourIE-GCN $^{type}$ ” for the instances in this category can be fixed if the model exploits the co-occurrence of event types and argument roles in the same sentences. In particular, for two event mentions with related event types in the same sentences, an entity mention that plays some role in one event tends to also play some related role in the other event. These co-occurrence can be captured via two event Argument edges and two entity Argument edges (sharing the same entity type) in the type dependency graphs of FourIE to address the issue. Consider an example sentence: “Two 13-year-old children were among those killed in the Haifa bus bombing, Israeli public radio said ...”. Both FourIE and “FourIE-GCN $^{type}$ ” can identify the Person entity mention “those” as the argument of role Victim for the Die event triggered by “killed”. However, regarding the Attack event triggered by “bombing”, only FourIE can correctly predict “those” as an argument of role Target. This success can be attributed to the ability of FourIE to learn the co-occurrence that an entity mention has a higher chance to play the role Target in an Attack event if it also has a role of Victim for a Die event mentioned in the same sentence.
Finally, the instances in the remaining  $15.4\%$  of  $\mathcal{A}$  tend to involve more complicated constraints/dependencies that cannot be associated with any of the three categories above.