In the Gaussian graphical models, the conditional dependencies among p variables can be represented by a graph $G = (V,E)$, where $V={1,2,...,p}$ is a set of nodes representing $p$ variables and $E = \{(i,j) | \omega_{ij} \ne 0, 1 \leq i \ne j \leq p\}$ is a set of edges corresponding to the nonzero off-diagonal elements of $\Omega$.

SPACE considers linear models such that for $i=1,2,...,p$, $$X_{i} = \sum_{j\neq i}\beta_{ij}X_{j} + \epsilon_{i}$$ where $\epsilon_{i}$ is an n-dimensional random vector from the multivariate normal distribution with mean $0$ and covariance matrix $1 / \omega_{ii}I_n$ is an identity matrix with size of $n×n$. Under normality, the regression coefficients $\beta_{ij}$ can be replaced with the partial correlations $\rho^{ij}$ by the relationship $$\beta_{ij} = - \frac{\omega_{ij}}{\omega_{ij}} = p^{ij}\sqrt{\frac{\omega_{jj}}{\omega_{ii}}}$$ where $p^{ij} = corr (X_{i}, X_{j} | X_{k}, k \neq i, j) = -\omega_{ij} /\sqrt{\omega_{ii}\omega_{jj}}$ is a partial correlation between $X_i$ and $X_j$. SPACE method solves the following $ℓ_1$-regularized problem: $$\underset{p}{min}\frac{1}{2}\sum_{i=1}^{p}\left \{ w_{i}\sum_{k=1}^{n} (X_{i}^{k} - \sum_{j\neq i}p^{ij}\sqrt{\frac{\omega_{ij}}{\omega_{ii}}}X_{j}^{k})^{2} \right \} + \lambda \sum_{1\leq i\le j \leq p}|p^{ij}|$$ where $w_i$ is a nonnegative weight for the $i$-th 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.
2. Peng, Jie, Pei Wang, Nengfeng Zhou, and Ji Zhu. "Partial correlation estimation by joint sparse regression models." Journal of the American Statistical Association 104.486 (2009): 735-746.