Abstract

We address the issue of dissipative vs. non-dissipative behavior in a mesoscopic set of coupled elements such as oscillators, with one half having gain and the other half having losses. We introduce a graph with coupling as the graph edges in given fixed number and gain/loss elements as its nodes. This relates to parity-time symmetry, notably in optics, e.g. set of coupled fibers, and more generally to the issue of taming divergence related to imaginary parts of eigenvectors in various network descriptions, for instance biochemical, neuronal, ecological. We thus look for the minimization of the imaginary part of all eigenvalues altogether, with a collective figure of merit. As more edges than gain/loss pairs are introduced, the unbroken cases , i.e., stable cases with real eigenvalues in spite of gain and loss, become statistically very scarce. A minimization from a random starting point by moving one edge at a time is studied, amounting to investigate how the hugely growing configuration number impedes the attainment of the desired minimally-dissipative target. The minimization path and its apparent stalling point are analyzed in terms of network connectivity metrics. We expand in the end on the relevance in biochemical signaling networks and the so-called “stability-optimized circuits” relevant to neural organization.

Graphical abstract

中文翻译：

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图上随机连接的介观集合对奇偶时间对称的配置障碍

摘要

我们在介观耦合元件（如振荡器）的介观集合中解决了耗散与非耗散行为的问题，其中一半具有增益，另一半具有损耗。我们引入一个图，该图在给定的固定数量下以耦合为图边缘，并以增益/损耗元素作为其节点。这涉及奇偶校验时间对称性，特别是在光学系统中，例如在一组耦合光纤中，并且更普遍地涉及与本征矢量的虚部相关的驯服散度的问题，这些特征涉及各种网络描述，例如生物化学，神经元，生态学。因此，我们希望所有特征值的虚部都最小化，并具有一个综合的品质因数。由于引入的边缘多于损益对，因此不间断的情况（即，尽管有损也有真实特征值的稳定情况）在统计上变得非常稀缺。研究了通过一次移动一个边缘从随机起点的最小化，从而研究了巨大增长的构型数如何阻碍达到所需的最小耗散目标。根据网络连接性指标分析了最小化路径及其明显的停滞点。最后，我们扩展了生化信号网络和与神经组织有关的所谓“稳定性优化电路”的相关性。

图形概要