The graphical lasso (GLASSO) method estimates a sparse inverse covariance matrix $\Omega$ by maximizing the $ℓ_1$ penalized log-likelihood $$l(\Omega) = log \left | \Omega \right | - tr(S\Omega) - \lambda \left \| \Omega \right \|_{1}$$ where $S$ is the sample covariance matrix, $tr(A)$ is the trace of $A$ and $∥A∥_1$ is the $ℓ_1$ norm of $A$ for $A \in \mathbb{R}^{p\times p}$.

To be specific, let $W$ be the estimate of the covariance matrix $\Sigma$ and consider partitioning $W$ and $S$ $$W = \binom{W_{11}\quad w_{12}}{w_{12}^{T}\quad w_{22}}, \quad S = \binom{S_{11}\quad s_{12}}{s_{12}^{T}\quad s_{22}},\quad \Omega = \binom{\Omega_{11}\quad \omega_{12}}{\omega_{12}^{T}\quad \omega_{22}} \quad (1)$$ The solution $\hat{\Omega}$ of $(1)$ is equivalent to the inverse of $W$ whose partitioned entity $w_{12}$ satisfies $w_{12}$ = $W_{11} \beta^{*}$ , where $\beta^{*}$ is the solution of the lasso problem $$\underset{\beta}{min} \frac{1}{2} \left\| W_{11}^{1/2}\beta - W_{11}^{-1/2}s_{12}\right\| _{2}^{2} + \lambda \left\| \beta \right\|_{1}.$$ Based on the above property, the graphical lasso sets the diagonal elements $w_{ii} = s_{ii} + \rho$ and obtains the off-diagonal elements of $W$ by repeatedly applying the following two steps:
     1. Permuting the columns and rows to locate the target elements at the position of $w_{12}$.
     2. Finding the solution $w_{12} = W_{11}\beta^*$ by solving the lasso problem. until convergence occurs. After finding $W$, the estimate $\hat{\Omega}$ is obtained from the relationship $\omega_{12} = -\hat{\beta}\hat{\omega}_{22}$ and $\hat{\omega}_{22} = 1/(\omega_{22} - \omega^T_{22}\hat{\beta})$, where $\hat{\beta} = W^{-1}_{11}w_{12}$.

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. Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. "Sparse inverse covariance estimation with the graphical lasso." Biostatistics 9.3 (2008): 432-441.

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Note:
Change the $\lambda$ value $(\lambda > 0)$ to control the sparsity of the network. The larger the $\lambda$ is, the more sparse the constructed network. If you don't know how to choose a value, use the default one.