论文标题
共同进化签名的异质网络的平均场限制
Mean field limits of co-evolutionary signed heterogeneous networks
论文作者
论文摘要
许多科学现象被建模为在静态网络上耦合的相互作用粒子系统（IP）。实际上，网络连接更具动态性。个人之间的联系会从附近的个人那里得到反馈，并进行更改以更好地适应世界。因此，合理的方法是将无数现实现象建模为共同进化（或自适应）网络是合理的。这些网络用于不同领域，包括电信，神经科学，计算机科学，生物化学，社会科学以及物理学，库拉莫托型网络已广泛用于模拟一组振荡器之间的交互。在本文中，我们提出了一种严格的公式，以实现在异质共进化网络上耦合的一系列共进化kuramoto振荡器的限制，这些振荡器从网络上振荡器的动力学中获得了积极和负面的反馈。我们在轻度条件下显示了共同进化网络的平均场极限（MFL），而共同进化的Kuramoto网络的序列会收敛到此MFL。这种MFL通过广义Vlasov方程的溶液描述。我们将图形限制视为签名的图测量，这是由[Kuehn，Xu的最新工作所激发的。 Digraph措施的Vlasov方程，JDE，339（2022），261--349]。与在非核进化网络上结合的IPS的最近新兴作品（即静态网络或与IPS动力学无关的时间依赖性网络）相比，我们的工作似乎是第一个严格地解决共同进化网络模型的MFL。
Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalized Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261--349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model.
