  We consider a model for chaotic diffusion with amplification on graphs
associated with piecewise-linear maps of the interval [S. Lepri, Chaos Solitons
& Fractals, 139,110003 (2020)]. We determine the conditions for having
fat-tailed invariant measures by considering approximate solution of the
Perron-Frobenius equation for generic graphs. An analogy with the statistical
mechanics of a directed polymer is presented that allows for a physically
appealing interpretation of the statistical regimes. The connection between
non-Gaussian statistics and the generalized Lyapunov exponents $L(q)$ is
illustrated. Finally, some results concerning large graphs are reported.
