In this paper we study the dynamic behavior of threshold networks on undirected signed graphs. While much attention has been given to the convergence and long-term behavior of this model, an open question remains: How does the underlying graph structure influence network dynamics? While similar papers have been carried out for threshold networks (as well as for other networks) these have largely focused on unsigned networks. However, the signed graph model finds applications in various real-world domains like gene regulation and social networks. By studying a graph parameter that we call "stability index," we search to establish a connection between the structure and the dynamics of threshold network. Interestingly, this parameter is related to the concepts of frustration and balance in signed graphs. We show that graphs that present negative stability index exhibit stable dynamics, meaning that the dynamics converges to fixed points regardless of threshold parameters. Conversely, if at least one subgraph has positive stability index, oscillations in long term behavior may appear. Finally, we generalize the analysis to network dynamics under periodic update schemes and we explore the case in which the stability index is positive for some subgraph finding that attractors with superpolynomial period on the size of the network may appear.

翻译：本文研究无向符号图上阈值网络的动态行为。尽管该模型的收敛性与长期行为已受到广泛关注，但仍存在一个开放性问题：底层图结构如何影响网络动力学？此前针对阈值网络（及其他网络）的类似研究主要集中于无符号网络。然而，符号图模型在基因调控、社交网络等现实领域具有广泛应用。通过研究我们称之为"稳定性指数"的图参数，试图建立阈值网络结构与动力学之间的关联。有趣的是，该参数与符号图中的挫折度与平衡性概念相关。研究表明，稳定性指数为负的图呈现稳定动力学特性，即无论阈值参数如何设置，动力学始终收敛至固定点。反之，若至少存在一个稳定性指数为正的子图，则可能出现长期振荡行为。最后，我们将分析推广至周期性更新方案下的网络动力学，并探讨当某些子图的稳定性指数为正时，发现可能出现周期长度关于网络规模超多项式的吸引子。