  Interdependent networks are more fragile under random attacks than simplex
networks, because interlayer dependencies lead to cascading failures and
finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning
that at the transition point the order parameter has a jump but there are also
critical phenomena related to it. Here we study these phenomena on the
Erd\H{o}s--R\'enyi and the two dimensional interdependent networks and show
that the hybrid percolation transition exhibits two kinds of critical
behaviors: divergence of the fluctuations of the order parameter and power-law
size distribution of finite avalanches at a transition point. At the transition
point, avalanches of infinite size occur thus the avalanche statistics also has
the nature of a HPT. The exponent $\beta_m$ of the order parameter is $1/2$
under general conditions, while the value of the exponent $\gamma_m$
characterizing the fluctuations of the order parameter depends on the system.
The critical behavior of the finite avalanches can be described by another set
of exponents, $\beta_a$ and $\gamma_a$. These two critical behaviors are
coupled by a scaling law: $1-\beta_m=\gamma_a$.
