Generalized priority-queue network dynamics: Impact of team and hierarchy

We study the effect of team and hierarchy on the waiting-time dynamics of priority-queue networks. To this end, we introduce generalized priority-queue network models incorporating interaction rules based on team-execution and hierarchy in decision making, respectively. It is numerically found that the waiting time distribution exhibits a power law for long waiting times in both cases, yet with different exponents depending on the team size and the position of queue nodes in the hierarchy, respectively. The observed power-law behaviors have in many cases a corresponding single or pairwise-interacting queue dynamics, suggesting that the pairwise interaction may constitute a major dynamics consequence in the priority-queue networks. It is also found that the reciprocity of influence is a relevant factor for the priority-queue network dynamics

💡 Research Summary

The paper investigates how two salient social structures—team collaboration and hierarchical decision‑making—affect the waiting‑time dynamics of priority‑queue networks. Traditional priority‑queue models assume that each node (representing an individual or agent) maintains a list of tasks with associated priorities, and at each time step the task with the highest priority is executed. Earlier studies have shown that, when only pairwise interactions are present, the distribution of waiting times τ follows a power‑law tail, (P(\tau)\sim \tau^{-\alpha}), indicating a high probability of very long delays. However, real‑world systems often involve groups of agents working together on a common task (team execution) or a hierarchy in which decisions made by higher‑level agents cascade down to subordinates. The authors therefore introduce two generalized interaction rules to capture these features.

Team‑execution rule: A task is assigned simultaneously to a fixed number (m) of nodes, forming a “team”. The task is removed from the system only after all (m) members have executed it. When (m=2) the rule reduces to the conventional pairwise interaction; for larger (m) the authors observe that the waiting‑time distribution still exhibits a power‑law tail, but the exponent (\alpha) decreases as the team size grows (e.g., (\alpha\approx1.5) for (m=3) and (\alpha\approx1.2) for (m=5)). This indicates that larger collaborative groups generate longer tails, reflecting the intuitive notion that coordination among many agents can delay overall progress.

Hierarchy rule: Each node is assigned a hierarchical level (h). When a node at a higher level selects a task, the same task is automatically imposed on all lower‑level nodes, whereas a lower‑level node’s choice does not affect higher‑level agents. This introduces a directed, non‑reciprocal influence. Simulations reveal that the waiting‑time exponent depends strongly on the node’s position: the top‑level node exhibits the smallest (\alpha) (≈1.1), producing the longest tail, while the bottom‑level node’s exponent approaches that of the pairwise model (≈1.7). The asymmetry of influence thus amplifies long‑delay events for agents high in the hierarchy.

A key finding is that, in the limiting cases where the team size is two or the hierarchy is flat (all nodes share the same level), the generalized models reproduce the same power‑law exponents as the classic pairwise model. This suggests that many complex interaction patterns can be effectively reduced to elementary pairwise dynamics, and that the pairwise interaction may be the dominant mechanism underlying the observed heavy‑tailed waiting times.

The authors also explore the role of reciprocity—whether influence is mutual. In the hierarchy rule, the lack of reciprocity leads to a steeper decay of the waiting‑time distribution, whereas the team rule, which inherently involves mutual influence among team members, sustains the heavy tail. Consequently, reciprocity emerges as a crucial factor for generating long‑range temporal correlations in the system.

From an applied perspective, the results have implications for project management, organizational design, and any domain where tasks are processed by interacting agents. Larger teams are predicted to experience more pronounced delays, while top‑down decision structures can exacerbate waiting times for high‑level agents, potentially creating bottlenecks. Recognizing that the power‑law tail persists across these variations underscores the inherent vulnerability of such systems to extreme delays, which is relevant for risk assessment and mitigation strategies.

The paper concludes by acknowledging limitations: the network topology is static, team composition does not evolve, and only a single fixed team size or hierarchy depth is examined at a time. Future work is suggested to incorporate dynamic rewiring, adaptive team formation, and empirical validation against real‑world datasets (e.g., email communication, collaborative software development logs). Overall, the study extends the theoretical foundation of priority‑queue networks by embedding realistic social structures, demonstrating that while the fundamental power‑law behavior remains robust, its quantitative characteristics are sensitively modulated by team size and hierarchical position.