A square matrix is said to be diagonally stable if there exists a diagonal matrix D > 0 satisfying DE + (ED)-D-T < 0. This notion has been instrumental in recent studies on stability of interconnected system models in communication and biological networks, in which the subsystems satisfy passivity properties and the matrix combines this passivity information with the interconnection structure. This paper presents a necessary and sufficient condition for diagonal stability when the digraph describing the network conforms to a "cactus" structure, which means that a pair of distinct simple circuits in the graph have at most one common vertex. In the special case of a single circuit, this diagonal stability test recovers the "secant criterion" that was recently derived for cyclic networks. The paper then incorporates the new diagonal stability test in network stability analysis where the diagonal entries of the matrix serve as weights in a Lyapunov function constructed from storage functions that verify passivity properties of the components. Finally, the paper illustrates this stability test on examples motivated by gene networks and population dynamics.