  We study how large functional networks can grow stably under possible
cascading overload failures and evaluated the maximum stable network size above
which even a small-scale failure would cause a fatal breakdown of the network.
Employing a model of cascading failures induced by temporally fluctuating
loads, the maximum stable size $n_{\text{max}}$ has been calculated as a
function of the load reduction parameter $r$ that characterizes how quickly the
total load is reduced during the cascade. If we reduce the total load
sufficiently fast ($r\ge r_{\text{c}}$), the network can grow infinitely.
Otherwise, $n_{\text{max}}$ is finite and increases with $r$. For a fixed
$r\,(<r_{\text{c}})$, $n_{\text{max}}$ for a scale-free network is larger than
that for an exponential network with the same average degree. We also discuss
how one detects and avoids the crisis of a fatal breakdown of the network from
the relation between the sizes of the initial network and the largest component
after an ordinarily occurring cascading failure.
