between $i$ and $j$ in the time interval $[0,t]$. For each pair $i,j$, the counting processes $N_{ij}, N_{ji}$ are mutually-exciting Hawkes processes with conditional intensities $$\begin{aligned}
\lambda_{ij}(t)=\mu_{ij} + \int_0^t g_\phi(t-u)\, dN_{ji}(u),\qquad \lambda_{ji}(t)=\mu_{ji} + \int_0^t g_\phi(t-u)\, dN_{ij}(u)
\label{eq:intensity}\end{aligned}$$ where $g_\phi$ is the exponential kernel defined in Equation . Interactions from individual $i$ to individual $j$ may arise as a response to past interactions from $j$ to $i$ through the kernel $g_\phi$, or via the base intensity $\mu_{ij}$. We also model assortativity so that individuals with similar interests are more likely to interact than individuals with different interests. For this, assume that each node $i$ has a set of positive latent parameters $(w_{i1},\ldots,w_{ip})\in \mathbb R_+^p$, where $w_{ik}$ is the level of its affiliation to each latent community $k=1,\ldots,p$. The number of communities $p$ is assumed known. We model the base rate $$\mu_{ij}=\mu_{ji}=\sum_{k=1}^p w_{ik}w_{jk}.\label{eq:muij}$$Two nodes with high levels of affiliation to the same communities will be more likely to interact than nodes with affiliation to different communities, favouring assortativity.

In order to capture sparsity and power-law properties and as in @Todeschini2016, the set of affiliation parameters $(w_{i1},\ldots,w_{ip})$ and node labels $\theta_i$ is modelled via a compound CRM with gamma scores, that is $W 