  Network filtering is an important form of dimension reduction to isolate the
core constituents of large and interconnected complex systems. We introduce a
new technique to filter large dimensional networks arising out of dynamical
behavior of the constituent nodes, exploiting their spectral properties. As
opposed to the well known network filters that rely on preserving key
topological properties of the realized network, our method treats the spectrum
as the fundamental object and preserves spectral properties. Applying
asymptotic theory for high dimensional data for the filter, we show that it can
be tuned to interpolate between zero filtering to maximal filtering that
induces sparsity and consistency while having the least spectral distance from
a linear shrinkage estimator. We apply our proposed filter to covariance
networks constructed from financial data, to extract the key subnetwork
embedded in the full sample network.
