cs.MAFeb 24, 2026

From Cooperation to Hierarchy: A Study of Dynamics of Hierarchy Emergence in a Multi-Agent System

Authors: Shanshan Mao, Peter Tino

Organizations: School of Computer Science, University of Birmingham, Birmingham, UK

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Abstract

A central premise in evolutionary biology is that individual variation can generate information asymmetries that facilitate the emergence of hierarchical organisation. To examine this process, we develop an agent-based model (ABM) to identify the minimal conditions under which hierarchy arises in dynamic multi-agent systems, focusing on the roles of initial heterogeneity and mutation amplitude across generations. Hierarchical organisation is quantified using the Trophic Incoherence (TI) metric, which captures directional asymmetries in interaction networks. Our results show that even small individual differences can be amplified through repeated local interactions involving reproduction, competition, and cooperation, but that hierarchical order is markedly more sensitive to mutation amplitude than to initial heterogeneity. Across repeated trials, stable hierarchies reliably emerge only when mutation amplitude is sufficiently high, while initial heterogeneity primarily affects early formation rather than long-term persistence. Overall, these findings demonstrate how simple interaction rules can give rise to both the emergence and persistence of hierarchical organisation, providing a quantitative account of how structured inequality can develop from initially homogeneous populations.

Figures & tables

Figure 1: Agent–environment interaction diagram adapted from [ 16 , 18 ] , illustrating the information flow between agent states, sensing, and action execution within the MAS.
SymbolDescriptionTypical Role
αti\alpha_{t}^{i}Individual abilitydynamic variable
Rti\mathcal{R}_{t}^{i}Percentile rank within team(0,1]
dts,dtld_{t}^{s},d_{t}^{l}Expected dominancespeaker/listener
dt,qd_{t,q}Consensus coefficientdynamic
rtir_{t}^{i}Resource share0–1
Table 1: Model parameters related to the consensus mechanism.
Figure 2: Mean Final Trophic Incoherence across parameter space. Each cell shows the mean final TI across 20 replications. Darker blue tones denote lower TI, indicating stronger hierarchical order.
Figure 3: Variability of TI across replications (IQR). The interquartile range captures cross-run variability for each (c,u)(c,u) pair. Smaller values indicate greater stability and convergence of hierarchical order.
Figure 4: Phase map of hierarchical regimes. Regions where median(TI)<0.45\text{median}(TI)<0.45 and IQR<0.05\text{IQR}<0.05 are classified as Consistent decrease (dark blue), indicating strong cross-run reproducibility across independent runs; transitional regions as Rebound (orange), and disordered regions as No-change (gray). The map highlights a clear boundary separating ordered and disordered regimes.
Figure 5: Representative trajectories at (c=0.05, u=1.0)(c=0.05,\,u=1.0) . Each gray line represents one of 20 runs; the blue curve is the median TI, and the shaded area the interquartile range (25–75%). The green region highlights periods satisfying the stability criterion ( median(TI)<0.45\text{median}(TI)<0.45 and IQR<0.05\text{IQR}<0.05 ), indicating the onset of a stable hierarchical regime.
Figure 6: Temporal dynamics of TI under varying mutation strengths uu . Each curve shows the median TI across 20 replicates (shaded area = IQR 25–75%); rows correspond to different initial heterogeneity cc . Higher uu values lead to earlier and deeper ordering, whereas low uu maintains high-TI, weakly ordered states.
Figure 7: Network Topologies at High and Low TI. (a) Communication network at step 30 000 with F(h) ⁣≈ ⁣0.99F(h)\!\approx\!0.99 ; (b) at step 30 000 with F(h) ⁣≈ ⁣0.31F(h)\!\approx\!0.31 . Both are visualized using a trophic-level layout positioning lower-level nodes at the bottom while minimizing edge crossings. Strong hierarchical ordering, as in (b), forms distinct tiers, whereas incoherent networks like (a) appear tangled and symmetric.
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