Dynamical Stability of Threshold Networks over Undirected Signed Graphs

TL;DR

This work analyzes how the structure of undirected signed graphs governs the dynamics of threshold networks by introducing the stability index $\mathcal{S}(G,w)$, built from the antifrustration index $\rho(G,w)$. It proves that negative stability on every induced subgraph guarantees convergence to fixed points under synchronous updates, while a nonnegative induced subgraph can enable oscillations, and it extends these results to periodic update schemes by requiring negativity of $\mathcal{S}$ on all induced subgraphs of each block. The results further extend to small-block periodic schemes, showing stability for blocks of size at most 2 and providing a catalog of larger graphs that can exhibit total-two cycles, with stability guaranteed if these subgraphs are avoided as induced subgraphs. The framework links antibalance structure to attractor patterns, offering structural criteria to predict dynamics in signed threshold networks, though computing $\rho(G,w)$ is NP-hard, suggesting future work on bounded-size update schemes and subgraph-based characterizations. This has potential implications for modeling regulatory and social networks where positive/negative interactions shape collective dynamics.

Abstract

This paper, we explore the dynamics of threshold networks on undirected signed graphs. Much attention has been dedicated to understanding 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 Boolean networks, but the latter primarily focus on unsigned networks. Here, we address this question in the context of signed threshold networks. We introduce the stability index of a signed graph, related to the concepts of antibalance in signed graphs. Our index establishes a connection between the structure and the dynamics of signed threshold networks. We show that signed graphs having a negative stability index on every induced subgraph exhibit stable dynamics, i.e., the dynamics converge to fixed points regardless of their threshold parameters. Conversely, if at least one induced subgraph has a non-negative stability index, oscillations in long-term behavior may appear. Furthermore, we generalize the analysis to network dynamics under periodic update schemes.

Figures (6)

• Figure 1: The asymptotic parallel (or synchronous) dynamics of two (stable) majority networks on some graphs in which the positive (resp. negative) edges are not labeled (resp are labeled by the minus symbol). In the upper panel, a majority network exhibiting a total limit cycle; in the lower panel, a majority network exhibiting a limit two-cycle. In this case, two of the nodes in the triangle are always in state $1$ and the rest of the nodes are switching.
• Figure 2: Example of a stable majority rule dynamics and of an unstable majority rule dynamics defined on the same graph (in which all the edges are positive and not explicitly labeled): on the left panel, the dynamics reaches a fixed point; on the right panel, the dynamics reaches a limit two-cycle.
• Figure 3: Threshold network based on the unstable majority rule whose associated graph is depicted on the upper panel. On the lower panel, three distinct evolution of configuration $(-1,1,-1,1)$ according to (a) the parallel update scheme, (b) the sequential update scheme $(\{1\},\{2\},\{4\},\{3\})$, (c) the block-sequential update scheme $(\{1,4\}, \{2,3\})$.
• Figure 4: Some examples of frustrated and non-frustrated cycles.
• Figure 5: Values for the stability index in different graphs. top-left: $\rho = 0$; top-right $\rho = 1$; bottom-left $\rho = 2$; bottom-right $\rho=1$.

Theorems & Definitions (30)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of \ref{['theo:totalcycles']}
• Remark 1
• Proposition 1: goles1980