  Growing networks decorated with antiferromagnetically coupled spins are
archetypal examples of complex systems due to the frustration and the
multivalley character of their energy landscapes. Here we use the damage
spreading method (DS) to investigate the cohesion of spin avalanches in the
exponential networks and the scale-free networks. On the contrary to the
conventional methods, the results obtained from DS suggest that the avalanche
spectra are characterized by the same statistics as the degree distribution in
their home networks. Further, the obtained mean range $Z$ of an avalanche, i.e.
the maximal distance reached by an avalanche from the damaged site, scales with
the avalanche size $s$ as $Z/N^\beta =f(s/N^{\alpha})$, where $\alpha=0.5$ and
$\beta=0.33$. These values are true for both kinds of networks for the number
$M$ of nodes to which new nodes are attached between 4 and 10; a check for M=25
confirms these values as well.
