lead to higher sparsity. For fixed $\alpha$, $$\begin{aligned}
\mathbb E[ V_{\alpha,T}]= \left\{
                                 \begin{array}{ll}
                                  \Theta( 1 ) & \text{if }\sigma<0 \\
                                   \Theta(\log T) & \text{if }\sigma=0 \\
                                  \Theta( T^{\sigma}) & \text{if }\sigma>0 \\
                                 \end{array}
                               \right.,\qquad
\mathbb E[ E_{\alpha,T}]= \left\{
                                 \begin{array}{ll}
                                  \Theta( 1 ) & \text{if }\sigma<0 \\
                                   O(\log T) & \text{if }\sigma=0 \\
                                  O( T^{3\sigma/2} ) & \text{if }\sigma\in(0,1/2] \\
                                  O( T^{(1+\sigma)/2}) & \text{if }\sigma\in(1/2,1) \\
                                 \end{array}
                               \right.\end{aligned}$$ and $\mathbb E[ I_{\alpha,T}]=\Theta(T )$ as $T$ tends to infinity. Sparsity in $T$ arises when $\sigma\geq 0$ for the number of edges and when $\sigma> 1/2$ for the number of interactions. The derivation of the asymptotic behaviour of expectations of $ V_{\alpha,T}$, $E_{\alpha,T}$ and $I_{\alpha,T}$ follows the lines of the proofs of Theorems 3 and 5.3 in [@Todeschini2016] $(\alpha\rightarrow\infty)$ and Lemma D.6 in the supplementary material of [@Cai2016] $(T\rightarrow\infty)$, and is omitted here.

Approximate Posterior Inference {#sec:inference}
===============================

Assume a set of observed interactions $\mathcal D=(t_k,i_k,j_k)_{k\geq 1}$ between $V$ individuals over a period of time $T$. The objective is to approximate the posterior distribution $\pi(\phi,\xi|\mathcal D)$ where $\phi$ are the kernel parameters and $\xi=((w_{ik})_{i=1,\ldots,V,k=1,\ldots,p},(a_k,b_k)_{k=1,\ldots,p},\alpha,\sigma,\tau)$, the parameters and hyperparameters of the compound CRM. One possible approach is to follow a similar approach to that 