Department of Physics and Interdisciplinary Center for Network Science & Applications, University of Notre Dame, Notre Dame, Indiana 46556, USA.
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA.
Phys Rev Lett. 2015 Apr 17;114(15):158701. doi: 10.1103/PhysRevLett.114.158701. Epub 2015 Apr 14.
Based on Jaynes's maximum entropy principle, exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data (observables). However, their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking, where predictions fail. Here we show that degeneracy appears when the corresponding density of states function is not log-concave, which is typically the consequence of nonlinear relationships between the constraining observables. Exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log-concave. The effectiveness of the method is demonstrated on examples.
基于 Jaynes 的最大熵原理，指数随机图为一系列原则性模型提供了基础，这些模型允许根据经验数据（可观测变量）预测网络属性。然而，它们的使用通常受到自发对称性破缺的退化问题的阻碍，在这种情况下，预测会失败。在这里，我们表明，当相应的态密度函数不是对数凹函数时，就会出现退化现象，这通常是约束可观测变量之间的非线性关系的结果。利用这些非线性关系，我们通过变换提出了一种解决一大类系统退化问题的方法，使态密度函数成为对数凹函数。该方法在示例中得到了验证。
