Convergence of time-delayed opinion dynamics with complex interaction types

TL;DR

This work analyzes convergence and convergence rate of time-delayed opinion dynamics on signed networks for both discrete-time and continuous-time settings. For discrete-time with delays $\tau_d$, it derives delay-robust convergence conditions that do not require all nodes to have self-loops and shows that larger delays typically slow the convergence via the leading eigenvalue $\theta_1$ of the augmented matrix $A$. For continuous-time with delays $\tau_c$, it establishes a feasible delay region bounded by a teardrop-shaped curve determined by the eigenvalues of $-L$, and proves that small delays can accelerate convergence while large delays can cause divergence; a Lambert $W$-based rate analysis further shows how delays can influence the convergence rate. Numerical simulations corroborate the theory across random and complex mixed interaction types, demonstrating the practical impact of delay and interaction topology on both convergence and rate.

Abstract

In opinion dynamics, time delays in agent-to-agent interactions are ubiquitous, which can substantially disrupt the dynamical processes rooted in agents' opinion exchange, decision-making, and feedback mechanisms. However, a thorough comprehension of quantitative impacts of time delays on the opinion evolution, considering diverse interaction types and system dynamics, remains absent. In this paper, we conduct a systematic investigation into the convergence and the associated rate of time-delayed opinion dynamics with diverse interaction types in both discrete-time and continuous-time systems. For the discrete-time system, we commence by establishing sufficient conditions for its convergence on arbitrary signed interaction networks. These conditions show that the convergence is determined solely by the topology of the interaction network and remains impervious to the magnitude of the time delay. Subsequently, we examine the influence of random and other interaction types on the convergence rate and discover that time delays tend to decelerate this rate. Regarding the continuous-time system, we derive the feasible domain of the delay that ensures the convergence of opinion dynamics, revealing that, unlike the discrete-time scenarios, large time delays can instigate the divergence of opinions. Specifically, we prove that for both random and other interaction types, small delays can expedite the convergence of continuous-time system. Finally, we present simulation examples to demonstrate the effectiveness and robustness of our research findings.

Figures (12)

• Figure 1: The interaction networks $\mathcal{G}(W)$ and $\mathcal{G}(A)$. Black lines and red lines represent trust and mistrust interactions, respectively.
• Figure 2: Three interaction networks without self-loops: $\mathcal{G}(W_{t})$, $\mathcal{G}(W_{m})$, and $\mathcal{G}(W_{tm})$. Black lines and red lines represent trust and mistrust interactions, respectively. In $(a)$, only trust interactions are present, while in $(b)$, only mistrust interactions are present. In $(c)$, both trust and mistrust interactions coexist.
• Figure 3: Convergence performances with evolution along the time axis for the discrete-time system (\ref{['equ2']}) associated with three interaction networks $W=W_{t}$, $W=W_{m}$, $W=W_{tm}$ in Fig. \ref{['pic-21']}, respectively.
• Figure 4: Convergence performances with evolution along the time axis for the continuous-time system (\ref{['equ1c1']}) associated with three interaction networks of Fig. 2, respectively.
• Figure 5: The boundary of the eigenvalue distribution of matrix $-L$ that ensures the convergence of system (\ref{['equ1c1']}). Solid lines and dots are obtained from our theory results and numerical simulations, respectively. Solid line denotes the curve C in Lemma 7 and dots are the eigenvalues of matrix $-L$.

Theorems & Definitions (40)

• Remark 1
• Lemma 1
• Proof 1
• Lemma 2
• Proof 2
• Lemma 3
• Proof 3
• Example 1
• Lemma 4
• Definition 1