We follow [@Rasmussen2013] and use vague Exponential$(0.01)$ priors on $\eta$ and $\delta$. Following @Todeschini2016 we set vague Gamma$(0.01, 0.01)$ priors on $\alpha$, $1-\sigma$, $\tau$, $a_k$ and $b_k$. The right limit for time window, $T$ is considered known.

Properties {#sec:properties}
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Connection to sparse vertex-exchangeable and edge-exchangeable models {#sec:properties2}
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The model is a natural extension of sparse vertex-exchangeable and edge-exchangeable graph models. Let $z_{ij}(t)=1_{N_{ij}(t)+N_{ji}(t)>0}$ be a binary variable indicating if there is at least one interaction in $[0,t]$ between nodes $i$ and $j$ in either direction. We assume $$\Pr(z_{ij}(t)=1|(w_{ik},w_{jk})_{k=1,\ldots,p})=1-e^{-2t\sum_{k=1}^p w_{ik}w_{jk}}$$ which corresponds to the probability of a connection in the static simple graph model proposed by @Todeschini2016. Additionally, for fixed $\alpha>0$ and $\eta=0$ (no reciprocal relationships), the model corresponds to a rank-$p$ extension of the rank-1 Poissonized version of edge-exchangeable models considered by @Cai2016 and @Janson2017. The sparsity properties of our model follow from the sparsity properties of these two classes of models.

Sparsity
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The size of the dataset is tuned by both $\alpha$ and $T$. Given these quantities, both the number of interactions and the number of nodes with at least one interaction are random variables. We now study the behaviour of these quantities, showing that the model 