The multivariate Lévy measure $\rho$ takes the form $$\rho(dw_{1},\ldots,dw_{p})=\int_{0}^{\infty
}w_{0}^{-p}\prod_{k=1}^p F_k\left(  \frac{dw_{k}}{w_{0}}\right)
\rho_{0}(dw_{0})
\label{eq:levymeasure}$$ where $F_k$ is the distribution of a Gamma random variable with parameters $a_k$ and $b_k$ and $\rho_0$ is the Lévy measure on $(0,\infty)$ of a generalized gamma process $$\rho_0(dw_0)=\frac{1}{\Gamma(1-\sigma)}w_0^{-1-\sigma}\exp(-w_0\tau)dw_0$$ where $\sigma\in(-\infty, 1)$ and $\tau>0$.

Denote ${\boldsymbol{w}}_i=(w_{i1},\dots,w_{ip})^T$, ${\boldsymbol{\beta}}_i=\left(\beta_{i1},\dots,\beta_{ip} \right)^T$ and $\rho(d{\boldsymbol{w}})=\rho(dw_1,\dots,dw_p)$. $w_0$ always refers to the scalar weight corresponding to the measure $\rho_0$.

B: Expected number of interactions, edges and nodes {#b-expected-number-of-interactions-edges-and-nodes .unnumbered}
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Recall that $I_{\alpha,T}, E_{\alpha,T}$ and $V_{\alpha,T}$ are respectively the overall number of interactions between nodes with label $\theta_i\leq \alpha$ until time $T$, the total number of pairs of nodes with label $\theta_i\leq \alpha$ who had at least one interaction before time $T$, and the number of nodes with label $\theta_i\leq \alpha$ who had at least one interaction before time $T$ respectively, and are defined as $$\begin{aligned}
I_{\alpha,T}&=\sum_{i\neq j}N_{ij}(T)1_{\theta_i\leq \alpha}1_{\theta_j\leq \alpha} \\
E_{\alpha,T}&=\sum_{i<j}1_{N_{ij}(T)+N_{ji}(T)>0}1_{\theta_i\leq \alpha}1_{\theta_j\leq \alpha}\\
V_{\alpha,T}&=\sum_{i}1_{\sum_{j\neq i} N_{ij}(T)1_{\theta_j\leq \alpha} >0}1_{\theta_i\leq \alpha}\end{aligned}$$

\[expected\_num\] The expected number of interactions $I_{\alpha,T}$, edges $E_{\alpha,T}$ and nodes $V_{\alpha,T}$ are given as follows: $$\begin{aligned}
\mathbb{E}[I_{\alpha,T}]{} & = \alpha^2 {\boldsymbol{\mu}}_w^T{\boldsymbol{\mu}}_w \left ( \frac{\delta}{\delta - \eta }T  -
      \frac{\eta}{(\eta -\delta)^2}  \left( 1 - e^{-T(\delta - \eta)}\right)\right )\\
\mathbb{E}[E_{\alpha,T}]&=
   \frac{\alpha^2}{2} \int_{\mathbb{R}^p_+} \psi(2Tw_1,\ldots,2Tw_p)  \rho(dw_{1},\ldots,dw_p)\\
