  In 2003, Ozsv\'ath and Szab\'o defined the concordance invariant $\tau$ for
knots in oriented 3-manifolds as part of the Heegaard Floer homology package.
In 2011, Sarkar gave a combinatorial definition of $\tau$ for knots in $S^3$
and a combinatorial proof that $\tau$ gives a lower bound for the slice genus
of a knot. Recently, Harvey and O'Donnol defined a relatively bigraded
combinatorial Heegaard Floer homology theory for transverse spatial graphs in
$S^3$ which extends knot Floer homology. We define a $\mathbb{Z}$-filtered
chain complex for balanced spatial graphs whose associated graded chain complex
has homology determined by Harvey and O'Donnol's graph Floer homology. We use
this to show that there is a well-defined $\tau$ invariant for balanced spatial
graphs generalizing the $\tau$ knot concordance invariant. In particular, this
defines a $\tau$ invariant for links in $S^3$. Using techniques similar to
those of Sarkar, we show that our $\tau$ invariant gives an obstruction to a
link being slice.
