  In this article, we present the first deterministic directed Laplacian L
systems solver that runs in time almost-linear in the number of non-zero
entries of L. Previous reductions imply the first deterministic almost-linear
time algorithms for computing various fundamental quantities on directed graphs
including stationary distributions, personalized PageRank, hitting times and
escape probabilities.
  We obtain these results by introducing partial symmetrization, a new
technique that makes the Laplacian of an Eulerian directed graph ``less
directed'' in a useful sense, which may be of independent interest. The
usefulness of this technique comes from two key observations: Firstly, the
partially symmetrized Laplacian preconditions the original Eulerian Laplacian
well in Richardson iteration, enabling us to construct a solver for the
original matrix from a solver for the partially symmetrized one. Secondly, the
undirected structure in the partially symmetrized Laplacian makes it possible
to sparsify the matrix very crudely, i.e. with large spectral error, and still
show that Richardson iterations convergence when using the sparsified matrix as
a preconditioner. This allows us to develop deterministic sparsification tools
for the partially symmetrized Laplacian.
  Together with previous reductions from directed Laplacians to Eulerian
Laplacians, our technique results in the first deterministic almost-linear time
algorithm for solving linear equations in directed Laplacians. To emphasize the
generality of our new technique, we show that two prominent existing
(randomized) frameworks for solving linear equations in Eulerian Laplacians can
be derandomized in this way: the squaring-based framework of Cohen, Kelner,
Peebles, Peng, Rao, Sidford and Vladu (STOC 2017) and the sparsified
Cholesky-based framework of Peng and Song (STOC 2022).
