Zhu Zheng, Andresen Juan Carlos, Moore M A, Katzgraber Helmut G
Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA.
Theoretische Physik, ETH Zurich, CH-8093 Zurich, Switzerland.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Feb;89(2):022118. doi: 10.1103/PhysRevE.89.022118. Epub 2014 Feb 18.
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.
我们研究了在外部偏置（磁场）下，具有竞争相互作用的无标度网络上布尔决策问题的平衡和非平衡性质。先前在零场下的研究表明，在无标度网络上具有竞争相互作用（自旋玻璃）的布尔变量（伊辛自旋）具有显著的平衡稳定性。当描述网络连通性幂律衰减的指数严格大于3时，系统会经历自旋玻璃转变。然而，当指数等于或小于3时，玻璃相在所有温度下都是稳定的。首先，我们在有场的情况下进行有限温度的蒙特卡罗模拟，以测试自旋玻璃相的稳健性，并表明系统在有场时具有自旋玻璃相，即表现出德阿尔梅达 - 图勒斯线。此外，我们研究了系统在零温度下由场驱动时的雪崩分布，以测试系统是否表现出自组织临界性。数值结果表明，当相互作用度的衰减指数小于或等于2时，雪崩（破坏）可以以非零概率在整个系统中传播，即具有竞争相互作用的无标度网络上的布尔决策问题在非热平衡时可能是脆弱的。
