one interaction between $i$ and $j$ in the interval $[0,T]$, and 0 otherwise.

For posterior inference, we employ an approximate procedure, which is formulated in two steps and is motivated by modular Bayesian inference (Jacob et al., 2017). It also gives another way to see the two natures of this type of temporal network data. Firstly we focus on the static graph i.e. the adjacency matrix of the pairs of interactions. Secondly, given the node pairs that have at least one interaction, we learn the rate for the appearance of those interactions assuming they appear in a reciprocal manner by mutual excitation.

We have $$\begin{aligned}
\pi(\phi,\xi|\mathcal D)&=\pi(\phi,\xi|\mathcal D,\mathcal Z)=\pi(\xi|\mathcal D,\mathcal Z)\pi(\phi|\xi ,\mathcal D).\end{aligned}$$ The idea of the two-step procedure is to

1.  Approximate $\pi(\xi|\mathcal D,\mathcal Z)$ by $\pi(\xi|\mathcal Z)$ and obtain a Bayesian point estimate $\widehat\xi$.

2.  Approximate $\pi(\phi|\xi ,\mathcal D)$ by $\pi(\phi|\widehat \xi ,\mathcal D)$.

C1: Stage 1 {#c1-stage-1 .unnumbered}
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As mentioned in Section \[sec:inference\] in the main article, the joint model $\pi(\mathcal Z, \xi)$ on the binary undirected graph is equivalent to the model introduced by [@Todeschini2016], and we will use their Markov chain Monte Carlo (MCMC) algorithm and the publicly available code SNetOC[^4] in order to 