How Leadership and Communication Delays Affect Consensus in Social Networks

Collective decision-making processes are prevalent in many real-world systems, ranging from social and economic networks to biological swarms and autonomous robotic groups. In these settings, individuals, or agents, continuously adjust their behavior or opinions based on information received from others in the population. In practice, however, these interactions are rarely instantaneous; information must be transmitted, processed, and interpreted before it can influence future actions. As a result, agents often react to outdated information, which inevitably introduces delays to the interaction process. These delays are unavoidable in many natural and engineered systems and can significantly affect the collective dynamics of the population.

Researchers have long used multi-agent models to describe this phenomenon in various areas of study, including opinion formation in social systems, synchronization in coupled oscillators, and alignment in collective motion. Classical frameworks such as the Hegselmann-Krause model [8], the Cucker-Smale model [5], and the Kuramoto model [9] successfully capture how local interactions can lead to global coordination. These models are motivated by their relevance in numerous scientific domains, including engineering, biology, and social sciences (see, for instance, [1]).

When studying opinion dynamics, one must account for another important feature: influential individuals or groups. These influential actors—typically political leaders, experts, or social media influencers—can shape the behavior of the broader population while remaining only partially influenced by it. The incorporation of such hierarchical interactions into mathematical models allows for a more realistic description of collective dynamics. In our work, we investigate how the combined effects of delayed communication and leader-follower interactions influence the emergence of consensus in large populations. In particular, we address the following question: Can a group of interacting individuals still reach agreement when information is delayed and influence is distributed hierarchically?

Mathematical Model

We investigate a time-delayed opinion formation model in which a population is divided into two groups: a small number of leaders and a larger group of followers, or non-leaders. Specifically, we consider \(m{\in}N\) leaders and \(N{\in}N\) non-leaders, with \(N>n\geq2.\)

What defines our setting is the asymmetric interaction structure; while leaders influence all agents in the system, they themselves are influenced only by other leaders. This structure models real-world situations where opinion leaders can shape the views of a population.

Let \(y_i(t){\in}R^d,i=1, ..., m\) denote the opinion of the \(i\)-th leader at time \(t,\) and \(x_i(t){\in}R^d, i=1, ..., N\) denote the opinion of the \(i\)-th follower. To account for the time required for discussion and decision-making, we introduced delays into the interactions among agents. The evolution of opinions is governed by a delayed Hegselmann-Krause-type system:

\[\frac{d}{dt}y_i(t)=\frac{1}{m}\sum^m_{j=1}{\psi}^{\tau_1}_{ij}(t)(y_j(t-\tau_1)-y_i(t)), \quad i=1, ..., m, \tag1\]

\[\frac{d}{dt}x_i(t)=\frac{1}{N}\sum^N_{j=1}{\phi}^{\tau_2}_{ij}(t)(x_j(t-\tau_2)-x_i(t)) + \frac{1}{m}\sum^m_{j=1}{\rho}^{\tau_1}_{ij}(t)(y_j(t-\tau_1)-x_i(t)) \quad i=1, ..., N. \tag2\]

The interaction weights are defined by

\[{\psi}^{\tau_1}_{ij}(t):={\psi}(y_i(t),y_j(t-\tau_1)),\quad {\phi}^{\tau_2}_{ij}(t):={\phi}(x_i(t), x_j(t-\tau_2)),\quad {\rho}^{\tau_1}_{ij}(t):={\rho}(x_i(t), y_j(t-\tau_1)),\]

where \({\psi},{\phi},\) and \(\rho\) are positive, bounded, and continuous interaction functions.

Our first main result concerns the convergence to consensus. If \(\{{y_i}(t)\}^m_{i=1}\) and \(\{{x_j}(t)\}^N_{j=1}\) denote the global-in-time solution of the system, then there exists a constant \({\gamma}>0,\) independent of \(m\) and \(N,\) such that the global diameter of the system decays exponentially:

\[d(t){\leq}e_{-{\gamma}(t-2\tau)}D_0.\]

Here the diameter \(d(t)\) represents the maximal distance between the opinions of any two agents. This means that all opinions eventually converge to the same value, so the entire population reaches agreement.

Large-scale Behavior: Mean-field Limit

To understand the collective behavior of systems with many interacting agents, we study the mean-field limit of the delayed particle system. We consider two different asymptotic regimes:

• Few Leaders and Many Followers: In the first regime, the number of leaders remains fixed while the number of followers tends to infinity. In this case, leaders retain their finite dimensional dynamics, with followers described by a probability density νt governed by a transport equation.
• Fully Macroscopic Regime: In the second regime, both leaders and followers become large populations and are described by probability densities. The resulting system consists of two coupled continuity equations describing the evolution of the leader and follower distributions.

In both cases, we prove existence and uniqueness of measure-valued solutions and show that the consensus property persists in the mean-field limit. In particular, the exponential decay of the diameter of the system is preserved at the macroscopic level.

Understanding the collective dynamics of large interacting systems is a central challenge in applied mathematics. Our analysis shows that consensus formation remains robust even in the presence of delayed communication and hierarchical interactions.

These results contribute to the mathematical understanding of opinion dynamics and multiagent systems and may have applications in areas such as social influence networks, distributed control, and collective decision-making processes.

Chiara Cicolani delivered a minisymposium presentation on this research at the 2025 SIAM Conference on Control and Its Applications (CT25), which took place in Montréal, Québec, Canada, last year. Cicolani attended CT25 through a Student Travel Award. To learn more about Student Travel Awards and submit an application, visit the online page.  

Acknowledgements: The author acknowledges collaborations with Young-Pil Choi and Cristina Pignotti on the mathematical results presented in this article.

References 
[1] Camazine, S., Deneubourg, J.L., Franks, N.R., Sneyd, J., Theraulaz, G., & Bonabeau, E. (2001). Self-organization in biological systems. Princeton, NJ: Princeton University Press. 
[2] Cicolani, C., Continelli, E., & Pignotti, C. (2025). First and second-order Cucker-Smale models with non-universal interaction, time delay, and communication failures, J. Dynam. Differential Equations. 
[3] Cicolani, C., Ouahab, B., & Pignotti, C. (2025). Opinion dynamics under common influencer assumption or leadership control. J. Optim. Theory Appl., 205, no. 5, p. 32.
[4] Cicolani, C. & Pignotti, C. (2025). Opinion dynamics of two populations with time-delayed coupling. Math. Methods Appl. Sci., 48(5), 5731–5744.
[5] Cucker, F. & Smale, S. (2007). Emergent behaviour in flocks. IEEE Trans. Autom. Control., 52, no. 5, 852–862.
[6] Halanay, A. (1966). Differential equations: Stability, oscillations, time lags. New York, NY:  Academic Press.
[7] Hale, J.K. & Lunel, S.M.V. (1993). Introduction to functional differential equations (Vol. 99,  Applied Mathematical Sciences). New York, NY: Springer.
[8] Hegselmann, R. & Krause, U. (2002). Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul., 5, no. 3, pp. 1–24.
[9] Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence (Vol. 19, Springer Series in Synergetics). Berlin, Germany: Springer-Verlag.