大动态系统在有向随机图上的线性稳定性分析
Linear stability analysis for large dynamical systems on directed random graphs
AI总结：
研究大动态系统在有向随机图上的线性稳定性，发现即使度分布有无限支撑，系统仍可能稳定；相变依赖于平均度和度相关系数等参数，且不稳定模式具有普遍特性。
摘要：
我们对定义在具有指定入度和出度分布的随机有向图上的大动态系统稳态（或固定点）进行了线性稳定性分析。我们得到两个重要结果：首先，即使度分布有无限支撑，无限大系统在有向图上仍可能稳定，这与无向图上当系统规模足够大时不稳定的结果形成对比。其次，稳定与不稳定相变是普遍的，仅依赖于少数参数，如平均度和度相关系数。此外，在不稳定区域，我们表征了破坏模式的性质，该模式也表现出普遍特性。这些结果源于对无限大局部树状且定向图的主特征值的精确理论，以及与主特征值相关的左右特征向量。我们通过无限大图的数值实验验证了分析结果。最后，我们讨论了该理论如何扩展到具有对角 disorder 的图以及包含无向链接的图，并讨论了小循环的影响，以及当它们诱导足够强的反馈回路时如何破坏大动态系统。
英文摘要：
We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally tree-like and oriented, as well as, for the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.
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