# Request for references: Best way to estimate directed graphs *with* cycles

I have $n$ observations of $p$ random variables $\{Y_1,\dots,Y_p \}$. I know that each observation has been generated from a linear network model of the form $$Y = WY+\epsilon$$ and that the graph described by $W$ is likely to be sparse and to have cycles. What is the current state of the art for estimating $W$? I have been reading about the graphical lasso but it seems limited to undirected graphs or acyclic directed graphs.

• Are $Y$ on the left hand side and $Y$ on the right hand side the same? Could you clarify what $Y$, $W$ and $\epsilon$ are? – Juho Kokkala Dec 8 '17 at 18:44
• Yes the $Y$ on the left and right hand side are the same. $W$ is a $p\times p$ matrix and $\epsilon$ is a vector of iid normal variable. – user_lambda Dec 8 '17 at 19:46
• So does this mean $Y_k = (I-W)^{-1}\epsilon_k$ or am I misunderstanding the setting completely? – Juho Kokkala Dec 8 '17 at 19:52
• Exactly (if $k$ is a vector of observations)! The issue is now to estimate $W$. – user_lambda Dec 8 '17 at 19:56
• Um, $k$ is an index(ing integer), $Y_k$ is a $p$-dimensional vector. – Juho Kokkala Dec 8 '17 at 20:04