This paper analyzes the convergence properties of signed networks with nonlinear edge functions. We consider diffusively coupled networks comprised of maximal equilibrium-independent passive (MEIP) dynamics on the nodes, and a general class of nonlinear coupling functions on the edges. The first contribution of this paper is to generalize the classical notion of signed networks for graphs with scalar weights to graphs with nonlinear edge functions using notions from the passivity theory. We show that the output of the network can finally form one or several steady-state clusters if all edges are positive and, in particular, all nodes can reach an output agreement if there is a connected subnetwork spanning all nodes and strictly positive edges. When there are nonpositive edges added to the network, we show that the tension of the network still converges to the equilibria of the edge functions if the relative outputs of the nodes connected by nonpositive edges converge to their equilibria. Furthermore, we establish the equivalent circuit models for signed nonlinear networks, and define the concept of equivalent edge functions, which is a generalization of the notion of the effective resistance. We finally characterize the relationship between the convergence property and the equivalent edge function, when a nonpositive edge is added to a strictly positive network comprised of nonlinear integrators. We show that the convergence of the network is always guaranteed, if the sum of the equivalent edge function of the previous network and the new edge function is passive.

中文翻译：

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有符号非线性网络的收敛性分析

本文分析了具有非线性边缘函数的有符号网络的收敛性。我们考虑由节点上的最大独立于平衡的无源（MEIP）动力学和边缘上的一类非线性耦合函数组成的扩散耦合网络。本文的第一个贡献是使用无源理论将具有标量权重的图的有符号网络的经典概念推广到具有非线性边缘函数的图。我们表明，如果所有边缘都是正的，则网络的输出最终可以形成一个或几个稳态簇，特别是如果存在跨越所有节点和严格呈正边缘的连接子网，则所有节点都可以达成输出协议。当网络上添加了非正边缘时，我们表明，如果由非正边缘连接的节点的相对输出收敛到其平衡，则网络的张力仍会收敛到边缘函数的平衡。此外，我们建立了有符号非线性网络的等效电路模型，并定义了等效边缘函数的概念，这是有效电阻概念的概括。当将非正边添加到由非线性积分器组成的严格正网络中时，我们最终描述了收敛性和等效边函数之间的关系。我们表明，如果先前网络的等效边缘函数与新边缘函数的和是被动的，则始终可以确保网络的收敛。