  We consider weighted directed networks for analysing, over the period
2000-2013, the interdependencies between volatilities of a large panel of
stocks belonging to the S\&P100 index. In particular, we focus on the so-called
{\it Long-Run Variance Decomposition Network} (LVDN), where the nodes are
stocks, and the weight associated with edge $(i,j)$ represents the proportion
of $h$-step-ahead forecast error variance of variable $i$ accounted for by
variable $j$'s innovations. To overcome the curse of dimensionality, we
decompose the panel into a component driven by few global, market-wide,
factors, and an idiosyncratic one modelled by means of a sparse vector
autoregression (VAR) model. Inversion of the VAR together with suitable
identification restrictions, produces the estimated network, by means of which
we can assess how {\it systemic} each firm is.~Our analysis demonstrates the
prominent role of financial firms as sources of contagion, especially during
the~2007-2008 crisis.
