  We study the equilibrium and nonequilibrium properties of Boolean decision
problems with competing interactions on scale-free networks in an external bias
(magnetic field). Previous studies at zero field have shown a remarkable
equilibrium stability of Boolean variables (Ising spins) with competing
interactions (spin glasses) on scale-free networks. When the exponent that
describes the power-law decay of the connectivity of the network is strictly
larger than 3, the system undergoes a spin-glass transition. However, when the
exponent is equal to or less than 3, the glass phase is stable for all
temperatures. First, we perform finite-temperature Monte Carlo simulations in a
field to test the robustness of the spin-glass phase and show that the system
has a spin-glass phase in a field, i.e., exhibits a de Almeida-Thouless line.
Furthermore, we study avalanche distributions when the system is driven by a
field at zero temperature to test if the system displays self-organized
criticality. Numerical results suggest that avalanches (damage) can spread
across the whole system with nonzero probability when the decay exponent of the
interaction degree is less than or equal to 2, i.e., that Boolean decision
problems on scale-free networks with competing interactions can be fragile when
not in thermal equilibrium.
