Chaotic fluctuations in graphs with amplification

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.