Boolean decision problems with competing interactions on scale-free networks: Equilibrium and nonequilibrium behavior in an external bias

摘要

We study the equilibrium and nonequilibrium properties of Boolean decisionproblems with competing interactions on scale-free networks in an external bias(magnetic field). Previous studies at zero field have shown a remarkableequilibrium stability of Boolean variables (Ising spins) with competinginteractions (spin glasses) on scale-free networks. When the exponent thatdescribes the power-law decay of the connectivity of the network is strictlylarger than 3, the system undergoes a spin-glass transition. However, when theexponent is equal to or less than 3, the glass phase is stable for alltemperatures. First, we perform finite-temperature Monte Carlo simulations in afield to test the robustness of the spin-glass phase and show that the systemhas 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 afield at zero temperature to test if the system displays self-organizedcriticality. Numerical results suggest that avalanches (damage) can spreadacross the whole system with nonzero probability when the decay exponent of theinteraction degree is less than or equal to 2, i.e., that Boolean decisionproblems on scale-free networks with competing interactions can be fragile whennot in thermal equilibrium.