  We present a method derived by cavity arguments to compute the spin-glass and
higher-order susceptibilities in diluted mean-field spin-glass models. The
divergence of the spin-glass susceptibility is associated to the existence of a
non-zero solution of a homogeneous linear integral equation. Higher order
susceptibilities, relevant for critical dynamics through the parameter exponent
$\lambda$, can be expressed at criticality as integrals involving the critical
eigenvector. The numerical evaluation of the corresponding analytic expressions
is discussed. The method is illustrated in the context of the de
Almeida-Thouless line for a spin-glass on a Bethe lattice but can be
generalized straightforwardly to more complex situations.
