parameter $\tau$ tunes the exponential cut-off in the degree distribution.[]{data-label="fig:degree"}](figures/tau_and_sigma_degreea "fig:"){width=".48\textwidth"}![Degree distribution for a Hawkes graph with different values of $\sigma$ (left) and $\tau$ (right). The degree of a node $i$ is defined as the number of nodes with whom it has at least one interaction. The value $\sigma$ tunes the slope of the degree distribution, larger values corresponding to a higher slope. The parameter $\tau$ tunes the exponential cut-off in the degree distribution.[]{data-label="fig:degree"}](figures/tau_and_sigma_degreeb "fig:"){width=".48\textwidth"}

The model admits the following set of hyperparameters, with the following interpretation:\
$\bullet$ The hyperparameters $\phi=(\eta,\delta)$ where $\eta\geq 0$ and $\delta\geq 0$ of the kernel $g_\phi$ tune the *reciprocity*.\
$\bullet$ The hyperparameters $(a_k,b_k)$ tune the *community structure* of the interactions. $a_k/b_k=\mathbb E[\beta_{ik}]$ tunes the size of community $k$ while $a_k/b_k^2=\text{var}(\beta_{ik})$ tunes the variability of the level of affiliation to this community; larger values imply more separated communities.\
$\bullet$ The hyperparameter $\sigma$ tunes the *sparsity* and the *degree heterogeneity*: larger values imply higher sparsity and heterogeneity. It also tunes the slope of the degree distribution. Parameter $\tau$ tunes the exponential cut-off in the degree distribution. This is illustrated in Figure \[fig:degree\].\
$\bullet$ Finally, the hyperparameters $\alpha$ and $T$ tune the overall number of interactions and nodes.

