properties mentioned above, which are often found in real world interactions. These are *sparsity*, *degree heterogeneity*, *community structure* and *reciprocity*. Combining all the properties in a single model is non trivial and there is no such construction to our knowledge. The proposed model generalises existing reciprocating relationships models [@Blundell2012] to the sparse and power-law regime. Our model can also be seen as a natural extension of the classes of models based on exchangeable random measures and edge-exchangeable models and it shares properties of both families. The approach is shown to outperform alternative models for link prediction on a variety of temporal network datasets.

The construction is based on Hawkes processes and the (static) model of @Todeschini2016 for sparse and modular graphs with overlapping community structure. In Section \[sec:background\], we present Hawkes processes and compound completely random measures which form the basis of our model’s construction. The statistical model for temporal dyadic data is presented in Section \[sec:model\] and its properties derived in Section \[sec:properties\]. The inference algorithm is described in Section \[sec:inference\]. Section \[sec:experiments\] presents experiments on four real-world temporal interaction datasets.

Background material {#sec:background}
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Hawkes processes
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Let $(t_k)_{k\geq 1}$ be a sequence of event times with $t_k\geq 