\mathbb{E}[V_{\alpha,T}]&=
\alpha  \int_{\mathbb{R}^p_+} \left (1-e^{- \alpha \psi(2Tw_1,\ldots,2Tw_p)}\right )\rho(dw_1,\ldots,dw_p)\end{aligned}$$ where $\mu_w =\int_{\mathbb{R}^p_+} {\boldsymbol{w}}\rho(dw_1,\ldots,dw_p)$.

The proof of Theorem \[expected\_num\] is given below and follows the lines of Theorem 3 in [@Todeschini2016].

Mean number of nodes $\mathbb{E}[V_{\alpha,T}]$ {#mean-number-of-nodes-mathbbev_alphat .unnumbered}
-----------------------------------------------

We have $$\begin{split}
\mathbb{E} \left[ V_{\alpha,T} \right] &=  \mathbb{E} \left[ \sum_i (1-1_{N_{ij}(T)=0, \forall j\neq i|\theta_j\leq \alpha)} \right]1_{\theta_i\leq \alpha} \nonumber\\
&= \sum_i \left\{ 1- \mathbb{P}(N_{ij}(T)=0, \forall j\neq i|\theta_j\leq \alpha) \right\}1_{\theta_i\leq \alpha}
\end{split}$$ Using the Palm/Slivnyak-Mecke formula and Campbell’s theorem, see e.g. [@Moller2003 Theorem 3.2] and [@Kingman1993], we obtain $$ \begin{split}
\mathbb{E}[V_{\alpha,T}]&=  \mathbb{E} \left[ \mathbb{E} \left[ V_{\alpha,T} |W_1,\ldots,W_p \right] \right] \\
&=  \mathbb{E} \left[\sum_i \left( 1-   e^{-2T{\boldsymbol{w}}_i^T \sum_{j\neq i} {\boldsymbol{w}}_j 1_{\theta_j \leq \alpha}  } \right) 1_{\theta_i \leq \alpha} \right]  \\
&= \alpha \int  \mathbb{E} \left( 1-   e^{-2T{\boldsymbol{w}}^T \sum_j {\boldsymbol{w}}_j 1_{\theta_j \leq \alpha}  } \right) \rho(d{\boldsymbol{w}}) \\
&= \alpha
\int_{\mathbb{R}_+^p} \left(1-e^{- \alpha \psi(2Tw_1,\ldots,2Tw_p)}\right) \rho \left(dw_1,\ldots,dw_p \right)
\end{split}
\label{nodes}$$

Mean number of edges $\mathbb{E}[E_{\alpha,T}]$ {#mean-number-of-edges-mathbbee_alphat .unnumbered}
-----------------------------------------------

Using the extended Slivnyak-Mecke formula, see e.g. [@Moller2003 Theorem 3.3] $$\begin{split}
 \mathbb{E}\left[E_{\alpha,T}\right]   &=\mathbb{E}\left[ \mathbb{E}\left[E_{\alpha,T}| W_1,\ldots,W_p \right] \right]  \nonumber\\
 & = \mathbb{E} \left[  \sum_i 1_{\theta_i<\alpha}\frac{1}{2} \sum_{j\neq i} 1_{\theta_j \leq \alpha}(1-e^{-2T{\boldsymbol{w}}_i^T{\boldsymbol{w}}_j})\right] \nonumber\\
   &= \frac{\alpha^2 }{2}\int_{\mathbb{R}^p_+} \psi(2Tw_1,\dots,2Tw_p)  \rho(dw_1,\ldots,dw_p)
\end{split}$$

Mean number of interactions $ \mathbb{E}[I_{\alpha,T}] $  {#mean-number-of-interactions-mathbbei_alphat .unnumbered}
---------------------------------------------------------

We have $$\begin{aligned}
 \mathbb{E}[I_{\alpha,T}]
