Stability and Robustness of Time-Varying Opinion Dynamics: A Graph-Theoretic Approach

TL;DR

Two temporal structures are introduced that serve as graph-theoretic certificates linking stubborn influence and temporal connectivity to contraction of the state-transition matrix, providing scalable and resilient tools for analyzing opinion formation in evolving social and human-AI networks.

Abstract

We study the stability of opinion dynamics in the time-varying Friedkin-Johnsen (TVFJ) model, which captures both persistent individual biases and adaptive social influence. We introduce two temporal structures, defected temporal graphs (DTGs) and weakly defected temporal graphs (WDTGs), that serve as graph-theoretic certificates linking stubborn influence and temporal connectivity to contraction of the state-transition matrix. Using these tools, we prove asymptotic stability of TVFJ dynamics under infinitely recurring DTGs, exponential stability in semi-periodic defected networks, and asymptotic stability of a trust-based extension under the weaker condition of recurring WDTGs. We also establish boundedness of the omega-limit set, showing that long-run opinions remain within the convex hull of innate beliefs, and characterize the limit set for periodically switching systems via a p-LTI decomposition with the tight bound that the size of the omega-limit set is at most p. Finally, we show that exponential stability persists under bounded perturbations, ensuring robustness in noisy or imperfect networks. These results unify algebraic contraction tests with interpretable graph-based reasoning, providing scalable and resilient tools for analyzing opinion formation in evolving social and human-AI networks.

Figures (6)

• Figure 1: Layer $\mathcal{G}(t_3)$ is a defected layer. All edges represent influential connections; red nodes indicate strictly stubborn agents, and green nodes are those temporally connected to the strictly stubborn agent via influential paths.
• Figure 2: Opinion trajectories of agents $v_1$ to $v_5$ under alternating networks with fixed Network 2 duration $d = 2$. The dynamics resemble French--DeGroot consensus due to increasing switching intervals.
• Figure 3: Interaction topology at time $t_k + d$, where $\mathcal{G}_{t_k + d}^{t_k + d + 1}$ forms a DTG. The presence of this DTG in every cycle guarantees asymptotic stability, regardless of initial conditions.
• Figure 4: Zero-input opinion trajectories under different initial conditions. All trajectories converge to zero, showing asymptotic stability and independence from initial conditions.
• Figure 5: Opinion trajectories of agents $\mathrm{v}_1$ to $\mathrm{v}_5$ in the periodic switching scenario with $\Lambda_2= \text{diag}(0,~0.9,~1,~1,~1)$.

Theorems & Definitions (37)

• Definition 1: Trust-Based Opinion Dynamics
• Remark 1
• Definition 2: Asymptotic and Exponential Stability
• Definition 3: Stubborn and Strictly Stubborn Agents
• Definition 4: Defected and Weakly Defected Temporal Graphs
• Remark 2
• Lemma 1
• proof
• Lemma 2
• proof