Consensus in networks of mobile communicating agents

Populations of mobile and communicating agents describe a vast array of technological and natural systems, ranging from sensor networks to animal groups. Here, we investigate how a group-level agreement may emerge in the continuously evolving network defined by the local interactions of the moving individuals. We adopt a general scheme of motion in two dimensions and we let the individuals interact through the minimal naming game, a prototypical scheme to investigate social consensus. We distinguish different regimes of convergence determined by the emission range of the agents and by their mobility, and we identify the corresponding scaling behaviors of the consensus time. In the same way, we rationalize also the behavior of the maximum memory used during the convergence process, which determines the minimum cognitive/storage capacity needed by the individuals. Overall, we believe that the simple and general model presented in this paper can represent a helpful reference for a better understanding of the behavior of populations of mobile agents.

💡 Research Summary

The paper investigates how a population of mobile agents, moving in a two‑dimensional square with periodic boundaries, reaches a global agreement when they interact through the minimal Naming Game (NG). Each agent moves at constant speed v, choosing a new random direction every τ_M time units, which yields a diffusive motion with diffusion coefficient D ∼ v²τ_M. Communication occurs via local broadcasting: at each discrete time step τ_S = 1 a randomly selected “speaker” emits a word to all agents within a Euclidean distance d. The speaker’s inventory is unchanged; receivers either delete all competing words and keep the transmitted one (if they already know it) or add the new word to their inventory. Initially all inventories are empty, so the first speaker invents a word.

The instantaneous communication network is a random geometric graph (RGG) defined by the emission range d. Its average degree is ⟨k⟩ = πNd²/L². The authors identify four characteristic ranges: d₁ (⟨k⟩ = 1), d_c (⟨k⟩ ≈ 4.51, the percolation threshold where a giant connected component (GCC) spans the system), d_{N/2} (⟨k⟩ = N/2, where a majority of agents receive each broadcast), and d_max (⟨k⟩ = N, a fully connected graph). By varying d and the agents’ speed v, three distinct dynamical regimes emerge, characterized by the ratio η = t₁/t_conv, where t₁ ∼ n(d)/v² is the typical time for an individual to leave a cluster of size n(d) and t_conv is the consensus time within that cluster.

1. Fast‑mixing regime (η ≪ 1, small d, large v). The network rewires faster than local consensus can develop. Agents constantly encounter new neighbors, leading to a global mixing process. Consensus time scales as t_conv ∝ 1/d² (or equivalently 1/⟨k⟩) because many broadcasts go unheard when ⟨k⟩ < 1. As d approaches d₁, t_conv follows this inverse‑square law.

2. Clustered regime (η ≫ 1, d < d_c, low v). Mobility is insufficient to dissolve small clusters; agents form isolated groups that quickly reach a local consensus on different words. Global agreement then emerges through competition between these locally agreed words. In this regime t_conv ∝ 1/v², reflecting that the limiting factor is the time required for clusters to exchange information rather than internal convergence.

3. Dense‑network regime (η ≫ 1, d ≫ d_c). The communication graph is already a GCC; the dynamics resemble those on static RGGs. Consensus time follows the known scaling for static geometric graphs, t_conv ∝ 1/d^{5/2}, until d exceeds d_{N/2}. For d > d_{N/2} the majority of agents hear each broadcast, making the process essentially instantaneous; the speed v becomes irrelevant.

Memory usage is quantified by M, the total number of distinct words present in the system at any time (equivalently the sum of all inventories). In static RGGs, larger ⟨k⟩ leads to higher M because more agents can store competing words simultaneously. In the mobile setting the same η‑dependence appears: the fast‑mixing regime (η ≪ 1) forces agents to encounter many different words, yielding a high M/N; the clustered regime (η ≫ 1, d ≈ d₁) minimizes M/N because each cluster quickly collapses to a single word; for intermediate d (between d_c and d_{N/2}) a peak in M/N occurs due to intense competition among several locally dominant words before the system finally converges.

The authors also explore the effect of population size N (keeping L fixed) and density ρ = N/L². Increasing N lowers the absolute values of d_c and d_{N/2} relative to the fixed box, so percolation and majority‑receiving regimes are reached at smaller d. For low mobility, t_conv saturates with N, while for high mobility it grows only weakly (logarithmically). The maximal memory per agent remains essentially constant across N, both for fast and slow agents.

Overall, the study provides a comprehensive scaling theory for consensus formation in time‑varying networks generated by mobile agents. By linking mobility, communication range, and population density to observable quantities (consensus time and memory demand), the work offers a useful reference for designing efficient protocols in robotic swarms, mobile sensor networks, and biological collectives where limited communication and storage resources are critical.