In this paper, we explore the dynamic behavior of threshold networks on undirected signed graphs. Much attention has been dedicated to understand the convergence and long-term behavior of this model. Yet, an open question persists: How does the underlying graph structure impact network dynamics? Similar studies have been carried out for threshold networks and other types of networks, but these primarily focus on unsigned networks. Here, we address the question on signed threshold networks. We introduce the stability index of a graph, related to the concepts of frustration and balance in signed graphs, to establish a connection between the structure and the dynamics of such networks. We show that graphs which present a negative stability index exhibit stable dynamics, i.e., the dynamics converges to fixed points regardless of its threshold parameters. Conversely, if at least one subgraph has a positive stability index, oscillations in long term behavior may appear. Furthermore, we generalize the analysis to network dynamics under periodic update modes and explore the case of the existence of some subgraph with a positive stability index, for which we find that attractors of super-polynomial period in the size of the network may appear.

翻译：本文探讨了无向符号图上阈值网络的动态行为。大量研究致力于理解该模型的收敛性和长期行为。然而，一个悬而未决的问题依然存在：底层图结构如何影响网络动力学？此前针对阈值网络及其他类型网络的研究已取得进展，但这些工作主要关注无符号网络。本文聚焦符号阈值网络问题，我们引入了图的稳定性指数（该概念与符号图中的挫折度和平衡性相关），以建立此类网络结构与动力学之间的联系。研究表明，具有负稳定性指数的图呈现稳定动力学特性，即无论阈值参数如何设置，其动力学均收敛至不动点。反之，若至少存在一个子图具有正稳定性指数，则长期行为可能出现振荡。此外，我们将分析推广至周期性更新模式下的网络动力学，并探究存在具有正稳定性指数的子图的情况，发现此时可能出现周期与网络规模呈超多项式关系的吸引子。