  We consider a model for chaotic diffusion with amplification on graphs
associated with piecewise-linear maps of the interval. We investigate the
possibility of having power-law tails in the invariant measure by approximate
solution of the Perron-Frobenius equation and discuss the connection with the
generalized Lyapunov exponents $L(q)$. We then consider the case of open maps
where trajectories escape and demonstrate that stationary power-law
distributions occur when $L(q)=r$, with $r$ being the escape rate. The proposed
system is a toy model for coupled active chaotic cavities or lasing networks
and allows to elucidate in a simple mathematical framework the conditions for
observing L\'evy statistical regimes and chaotic intermittency in such systems.
