通过矩闭包保持分岔
Preserving Bifurcations through Moment Closures
AI总结：
本文提出一种以定性特征（如分岔）为导向的矩闭包选择方法，严格推导了SIS流行病和自适应投票者模型中闭包需满足的条件，以实现跨临界分岔的保持。
摘要：
矩系统出现在广泛的情境和应用中，例如复杂系统的网络建模。由于矩系统由大量甚至无限个耦合方程组成，在许多情况下，获得适合进一步分析的低维表示的一个不可或缺的步骤是选择矩闭包。矩闭包由一组近似组成，这些近似将某些高阶矩用低阶矩来表示，从而应用这些近似后得到一个仅涉及低阶矩的封闭方程组。闭包通常凭借直觉和启发式方法来寻找，以得到定量上良好的近似。与此相反，我们提出了一种替代方法，转而关注能够产生某些定性特征（如分岔）的闭包。重要的是，这种根本性的视角转变为根据这些特征严格分类矩闭包提供了可能性。这使得闭包的设计和选择更加算法化、精确和可靠。在这项工作中，我们仔细研究了在两种广为人知的网络动力系统的平均场描述中出现的矩系统，即SIS流行病和自适应投票者模型。我们推导出任何矩闭包必须满足的条件，以便相应的封闭系统表现出这些系统中从随机粒子模型预期会出现的跨临界分岔。
英文摘要：
Moment systems arise in a wide range of contexts and applications, e.g. in network modeling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step in obtaining a low-dimensional representation that is amenable to further analysis is, in many cases, to select a moment closure. A moment closure consists of a set of approximations that express certain higher-order moments in terms of lower-order ones, so that applying those leads to a closed system of equations for only the lower-order moments. Closures are frequently found drawing on intuition and heuristics to come up with quantitatively good approximations. In contrast to that, we propose an alternative approach where we instead focus on closures giving rise to certain qualitative features, such as bifurcations. Importantly, this fundamental change of perspective provides one with the possibility of classifying moment closures rigorously in regard to these features. This makes the design and selection of closures more algorithmic, precise, and reliable. In this work, we carefully study the moment systems that arise in the mean-field descriptions of two widely known network dynamical systems, the SIS epidemic and the adaptive voter model. We derive conditions that any moment closure has to satisfy so that the corresponding closed systems exhibit the transcritical bifurcation that one expects in these systems coming from the stochastic particle model.
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