  We study how the recurrence and transience of space-time sets for a branching
random walk on a graph depends on the offspring distribution. Here, we say that
a space-time set $A$ is recurrent if it is visited infinitely often almost
surely on the event that the branching random walk survives forever, and say
that $A$ is transient if it is visited at most finitely often almost surely. We
prove that if $\mu$ and $\nu$ are supercritical offspring distributions with
means $\bar \mu < \bar \nu$ then every space-time set that is recurrent with
respect to the offspring distribution $\mu$ is also recurrent with respect to
the offspring distribution $\nu$ and similarly that every space-time set that
is transient with respect to the offspring distribution $\nu$ is also transient
with respect to the offspring distribution $\mu$. To prove this, we introduce a
new order on probability measures that we call the germ order and prove more
generally that the same result holds whenever $\mu$ is smaller than $\nu$ in
the germ order. Our work is inspired by the work of Johnson and Junge (AIHP
2018), who used related stochastic orders to study the frog model.
