with mean measure $\rho_0(dw_0)H_0(d\theta)$. In the rest of this paper, we assume that $F_k$ is a gamma distribution with parameters $(a_k,b_k)$, $H_0(d\theta)=d\theta$ is the Lebesgue measure and $\rho_0$ is the Lévy measure of a generalized gamma process $$\begin{aligned}
\rho_0(dw)=\frac{1}{\Gamma(1-\sigma)}w^{-1-\sigma}e^{-\tau w}dw\label{eq:levyGGP}\end{aligned}$$ where $\sigma\in(-\infty, 1)$ and $\tau>0$.

Statistical model for temporal interaction data {#sec:model}
===============================================

Consider temporal interaction data of the form $\mathcal D=(t_k,i_k,j_k)_{k\geq 1}$ where $(t_k,i_k,j_k)\in \mathbb R_+\times \mathbb N_*^2$ represents a directed interaction at time $t_k$ from node/individual $i_k$ to node/individual $j_k$. For example, the data may correspond to the exchange of messages between students on an online social network.

We use a point process $(t_k,U_k,V_k)_{k\geq 1}$ on $\mathbb R^3_+$, and consider that each node $i$ is assigned some continuous label $\theta_i\geq 0$. the labels are only used for the model construction, similarly to [@Caron2017; @Todeschini2016], and are not observed nor inferred from the data. A point at location $(t_k,U_k,V_k)$ indicates that there is a directed interaction between the nodes $U_k$ and $V_k$ at time $t_k$. See Figure \[fig:model\] for an illustration.

For a pair of nodes $i$ and $j$, with labels $\theta_i$ and $\theta_j$, let $N_{ij}(t)$ be the counting process $$\begin{aligned}
N_{ij}(t)=\sum_{k|(U_k,V_k)=(\theta_i,\theta_j)}1_{t_k\leq t}
\label{eq:PointProcess}\end{aligned}$$ for the number of interactions 