  We study the time-reversal dynamics of a tight-binding electron in the
Watts-Strogatz (WS) small-world networks. The localized initial wave packet at
time $t=0$ diffuses as time proceeds until the time-reversal operation,
together with the momentum perturbation of the strength $\eta$, is made at the
reversal time $T$. The time irreversibility is measured by $I \equiv |\Pi(t =
2T) - \Pi(t = 0)|$, where $\Pi$ is the participation ratio gauging the
extendedness of the wavefunction and for convenience, $t$ is measured forward
even after the time reversal . When $\eta = 0$, the time evolution after $T$
makes the wavefunction at $t=2T$ identical to the one at $t=0$, and we find
I=0, implying a null irreversibility or a complete reversibility. On the other
hand, as $\eta$ is increased from zero, the reversibility becomes weaker, and
we observe enhancement of the irreversibility. We find that $I$ linearly
increases with increasing $\eta$ in the weakly-perturbed region, and that the
irreversibility is much stronger in the WS network than in the local regular
network.
