From equation (7) in the cited paper , the Lieb-Robinson bound for the commutator of local operators $W_x$ and $W_y$, where the subscript indicates the location, is
The preceding pages defines the "radius" of an operator $W_x$ to be "the region in which [other] local operators have an order-one commutator with [the given operator]." In other words, the radius of $W_x(t)$ is the largest value of $|x-y|$ for which the commutator $[W_x(t),\,W_y]$ is of order $1$. The bound (7) says that if $|x-y|$ is much larger than $(c_1/c_2)t$, then the magnitude of the commutator is exponentially suppressed (assuming $c_2>0$). Therefore, the radius is (softly) bounded by $(c_1/c_2)t$, which means that it cannot grow faster than linearly-in-time.
 Roberts et al (2014), "Localized shocks," https://arxiv.org/abs/1409.8180