  We investigate the effect of disorder on the Curie-Weiss model with Glauber
dynamics. In particular, we study metastability for spin-flip dynamics on the
Erd\H{o}s-R\'enyi random graph $ER_n(p)$ with $n$ vertices and with edge
retention probability $p \in (0,1)$. Each vertex carries an Ising spin that can
take the values $-1$ or $+1$. Single spins interact with an external magnetic
field $h \in (0,\infty)$, while pairs of spins at vertices connected by an edge
interact with each other with ferromagnetic interaction strength $1/n$. Spins
flip according to a Metropolis dynamics at inverse temperature $\beta$. The
standard Curie-Weiss model corresponds to the case $p=1$, because $ER_n(1) =
K_n$ is the complete graph on $n$ vertices. For $\beta>\beta_c$ and $h \in (0,p
\chi(\beta p))$ the system exhibits \emph{metastable behaviour} in the limit as
$n\to\infty$, where $\beta_c=1/p$ is the \emph{critical inverse temperature}
and $\chi$ is a certain \emph{threshold function} satisfying
$\lim_{\lambda\to\infty} \chi(\lambda) =1$ and $\lim_{\lambda \downarrow 1}
\chi(\lambda)=0$. We compute the average crossover time from the
\emph{metastable set} (with magnetization corresponding to the `minus-phase')
to the \emph{stable set} (with magnetization corresponding to the
`plus-phase'). We show that the average crossover time grows exponentially fast
with $n$, with an exponent that is the same as for the Curie-Weiss model with
external magnetic field $h$ and with ferromagnetic interaction strength $p/n$.
We show that the correction term to the exponential asymptotics is a
multiplicative error term that is \emph{at most polynomial} in $n$. For the
complete graph $K_n$ the correction term is known to be a multiplicative
constant.
