To incorporate information about hub nodes, ESPACE extends the SPACE model by using an additional tuning parameter $\alpha$ on edges connected to the given hub nodes. $\lambda$ is the lasso penalty term. $\alpha$ reflect the hub gene information by reducing the penalty on edges connected to hub nodes. To be specific, let $H$ be the set of hub nodes that were previously identified. The ESPACE method we propose solves $$\underset{p}{min}\frac{1}{2}\sum_{i=1}^{p}\left \{ w_{i}\sum_{k=1}^{n} (X_{i}^{k} - \sum_{j\neq i}p^{ij}\sqrt{\frac{\omega_{ij}}{\omega_{ii}}}X_{j}^{k})^{2} \right \} + \alpha\lambda \sum_{i < j, \left \{ i\in H\right \}\cup \left \{ j\in H\right \}}|p^{ij}| + \lambda \sum_{i < j, i,j\in H^{c}}|p^{ij}|,$$ where $\lambda \geq 0$, $0 \leq \alpha \leq 1$. $w_i$ is weighted for the squared error loss.

Reference:
1. Donghyeon Yu, Johan Lim, Xinlei Wang, Faming Liang, and Guanghua Xiao. "Enhanced construction of gene regulatory networks using hub gene information." BMC bioinformatics 18.1 (2017): 186.

Note:
1. Change the $\lambda$ value $(\lambda \geq 0)$ to control the sparsity of the network. A larger $\lambda$ will give you a more sparse network. If you don't know how to choose a value, use the default one.
2. Change the $\alpha$ value $(0 \leq \alpha \leq 1)$ to control the penalty on hub genes. A smaller $\alpha$ will give less penalty on edges connected to hub genes. If you don't know how to choose a value, use the default one.
3. The hub gene input should be gene names separated by a comma, e.g. "Gene13,Gene52,Gene199". All the gene names must be contained in column names of the expression data.