  We deduce and discuss the implications of self-similarity for the stability
in terms of robustness to failure of multiplexes, depending on interlayer
degree correlations. First, we define self-similarity of multiplexes and we
illustrate the concept in practice using the configuration model ensemble.
Circumscribing robustness to survival of the mutually percolated state, we find
a new explanation based on self-similarity both for the observed fragility of
interconnected systems of networks and for their robustness to failure when
interlayer degree correlations are present. Extending the self-similarity
arguments, we show that interlayer degree correlations can change completely
the stability properties of self-similar multiplexes, so that they can even
recover a zero percolation threshold and a continuous transition in the
thermodynamic limit, qualitatively exhibiting thus the ordinary stability
attributes of noninteracting networks. We confirm these results with numerical
simulations.
