We introduce a novel two-component random network. Unit resistors are placed at random along the bonds of a pure superconducting linear chain, with the distance $l$ between successive resistors being chosen from the distribution $P(l)\ensuremath{\sim}{l}^{\ensuremath{-}(\ensuremath{\alpha}+1)}$ where $\ensuremath{\alpha}g0$ is a tunable parameter. We study the transport exponents ${d}_{w}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ defined by $〈{x}^{2}〉{t}^{\frac{2}{{d}_{w}}}$ and $\ensuremath{\rho}\ensuremath{\sim}{L}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}}$, where $〈{x}^{2}〉$ is the mean-square displacement, $\ensuremath{\rho}$ the resistivity, and $L$ the system size. We find that for $\ensuremath{\alpha}\ensuremath{\ge}1$ both ${d}_{w}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ stick at their value for a nonzero concentration of resistors. For $\ensuremath{\alpha}l1$ they vary continuously with $\ensuremath{\alpha}: {d}_{w}=2\ensuremath{\alpha}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}=\ensuremath{\alpha}$. In the presence of a bias field, we find ${d}_{w}=\ensuremath{\alpha}$. This is the first exactly soluble model displaying "anomalous ballistic diffusion," which we interpret physically in terms of a L\'evy-flight-type random walk on a linear chain lattice.