_0 =\sum_{i=1}^{\infty}w_{i0} \delta_{\theta_{i}} \sim\text{CRM} (\rho_{0})$ where the Lévy measure $\rho_0$ is defined by Equation , and for each node $i\geq 1$ and community $k=1,\ldots,p$ $$\begin{aligned}
w_{ik}= w_{i0}  \beta_{ik}\text{, where }\beta_{ik} \overset{\text{ind}}{\sim}\operatorname{Gamma}(a_k,b_k).
\label{weights}\end{aligned}$$ The parameter $w_{i0}\geq 0$ is a degree correction for node $i$ and can be interpreted as measuring the overall popularity/sociability of a given node $i$ irrespective of its level of affiliation to the different communities. An individual $i$ with a high sociability parameter $w_{i0}$ will be more likely to have interactions overall than individuals with low sociability parameters. The scores $\beta_{ik}$ tune the level of affiliation of individual $i$ to the community $k$. The model is defined on $\mathbb R_+^3$. We assume that we observe interactions over a subset $[0,T]\times[0,\alpha]^2\subseteq \mathbb R_+^3$ where $\alpha$ and $T$ tune both the number of nodes and number of interactions. The whole model is illustrated in Figure \[fig:model\].

![Degree distribution for a Hawkes graph with different values of $\sigma$ (left) and $\tau$ (right). The degree of a node $i$ is defined as the number of nodes with whom it has at least one interaction. The value $\sigma$ tunes the slope of the degree distribution, larger values corresponding to a higher slope. The 