  We instigate the properties of the threshold contact process (TCP), a process
showing an absorbing-state phase transition with infinitely many absorbing
states, on random complex networks. The finite size scaling exponents
characterizing the transition are obtained in a heterogeneous mean field (HMF)
approximation and compared with extensive simulations, particularly in the case
of heterogeneous scale-free networks. We observe that the TCP exhibits the same
critical properties as the contact process (CP), which undergoes an
absorbing-state phase transition to a single absorbing state. The accordance
among the critical exponents of different models and networks leads to
conjecture that the critical behavior of the contact process in a HMF theory is
a universal feature of absorbing state phase transitions in complex networks,
depending only on the locality of the interactions and independent of the
number of absorbing states. The conditions for the applicability of the
conjecture are discussed considering a parallel with the
susceptible-infected-susceptible epidemic spreading model, which in fact
belongs to a different universality class in complex networks.
