Dynamical Stability of Threshold Networks over Undirected Signed Graphs

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.