Abstract We aim to provide a theoretical framework to explain the discrete transitions of mood connecting ideas from network theory and dynamical systems theory. It was recently shown how networks (graphs) can be used to represent psychopathologies, where symptoms of, say, depression, affect each other and certain configurations determine whether someone could transition into a depression. To analyse changes over time and characterise possible future behaviour is in general rather difficult for large graphs. We describe the dynamics of graphs using one-dimensional discrete time dynamical systems theory obtained from a mean field approximation to stochastic cellular automata (SCA). Often the mean field approximation is used on a regular graph (a grid or torus) where each node has the same number of edges and the same probability of becoming active. We provide quantitative results on the accuracy of using the mean field approximation for the grid and random and small-world graph to describe the dynamics of the SCA. Bifurcation diagrams for the mean field of the different graphs indicate possible phase transitions for certain parameter settings of the mean field. Simulations confirm for different graph sizes (number of nodes) that the mean field approximation is accurate.

中文翻译：

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随机和小世界图的随机元胞自动机的平均场动力学

摘要 我们旨在提供一个理论框架来解释网络理论和动力系统理论中连接思想的情绪的离散转变。最近展示了如何使用网络（图表）来表示精神病理学，例如抑郁症的症状会相互影响，某些配置决定了某人是否会转变为抑郁症。分析随时间的变化并表征可能的未来行为对于大图来说通常是相当困难的。我们使用从平均场近似到随机元胞自动机 (SCA) 获得的一维离散时间动力系统理论来描述图形的动力学。通常平均场近似用于正则图（网格或环面），其中每个节点具有相同数量的边和相同的激活概率。我们提供了关于使用网格和随机和小世界图的平均场近似来描述 SCA 动态的准确性的定量结果。不同图的平均场的分叉图表示平均场的某些参数设置的可能相变。模拟确认对于不同的图形大小（节点数），平均场近似是准确的。