exhibits sparsity. Let $I_{\alpha,T}, E_{\alpha,T}, V_{\alpha,T}$ be 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.

$$\begin{aligned}
\begin{split}
V_{\alpha,T}&=\sum_{i}1_{\sum_{j\neq i} N_{ij}(T)1_{\theta_j\leq \alpha} >0}1_{\theta_i\leq \alpha}
\qquad I_{\alpha,T}=\sum_{i\neq j}N_{ij}(T)1_{\theta_i\leq \alpha}1_{\theta_j\leq \alpha} \\
E_{\alpha,T}&=\sum_{i<j}1_{N_{ij}(T)+N_{ji}(T)>0}1_{\theta_i\leq \alpha}1_{\theta_j\leq \alpha}
\end{split}\end{aligned}$$

We provide in the supplementary material a theorem for the exact expectations of $I_{\alpha,T}$, $E_{\alpha,T}$ $V_{\alpha,T}$. Now consider the asymptotic behaviour of the expectations of $ V_{\alpha,T}$, $E_{\alpha,T}$ and $I_{\alpha,T}$ , as $\alpha$ and $T$ go to infinity.[^1] Consider fixed $T>0$ and $\alpha$ that tends to infinity. Then, $$\begin{aligned}
\mathbb E[ V_{\alpha,T}]= \left\{
                                 \begin{array}{ll}
                                  \Theta( \alpha) & \text{if }\sigma<0 \\
                                   \omega(\alpha) & \text{if }\sigma\geq 0
                                 \end{array}
                               \right.,\qquad
\mathbb E[ E_{\alpha,T}]=\Theta(\alpha^2),\qquad
\mathbb E[ I_{\alpha,T}]=\Theta(\alpha^2)\end{aligned}$$ as $\alpha$ tends to infinity. For $\sigma<0$, the number of edges and interactions grows quadratically with the number of nodes, and we are in the dense regime. When $\sigma\geq 0$, the number of edges and interaction grows subquadratically, and we are in the sparse regime. Higher values of $\sigma$ 