  The Lieb-Robinson bound sets a theoretical upper limit on the speed at which
information can propagate in non-relativistic quantum spin networks. In its
original version, it results in an exponentially exploding function of the
evolution time, which is partially mitigated by an exponentially decreasing
term that instead depends upon the distance covered by the signal (the ratio
between the two exponents effectively defining an upper bound on the
propagation speed). In the present paper, by properly accounting for the free
parameters of the model, we show how to turn this construction into a stronger
inequality where the upper limit only scales polynomially with respect to the
evolution time. Our analysis applies to any chosen topology of the network, as
long as the range of the associated interaction is explicitly finite. For the
special case of linear spin networks we present also an alternative derivation
based on a perturbative expansion approach which improves the previous
inequality. In the same context we also establish a lower bound to the speed of
the information spread which yields a non trivial result at least in the limit
of small propagation times.
