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arxiv: 2602.21404 · v2 · submitted 2026-02-24 · 💻 cs.MA

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

Shanshan Mao , Peter Tino This is my paper

Pith reviewed 2026-05-15 19:23 UTC · model grok-4.3

classification 💻 cs.MA
keywords multi-agent systemshierarchy emergenceagent-based modelingtrophic incoherencemutation amplitudecooperationcompetitionreproduction
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The pith

Stable hierarchies emerge reliably in multi-agent systems only when mutation amplitude is high enough, with initial differences mattering mainly for early formation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds an agent-based model to test how hierarchy arises from individual variation through local interactions. It demonstrates that rules for reproduction, competition, and cooperation can amplify small differences into directional network asymmetries. Hierarchy measured by trophic incoherence persists stably only above a mutation amplitude threshold, while starting heterogeneity speeds early stages but fades over time. This shows structured inequality can develop from nearly uniform populations using minimal interaction rules.

Core claim

In the agent-based model, hierarchical organisation quantified by the trophic incoherence metric emerges and persists across trials when mutation amplitude is sufficiently high; initial heterogeneity influences early formation but has little effect on long-term stability. Simple rules of reproduction, competition, and cooperation suffice to generate directional asymmetries in interaction networks from initially homogeneous populations.

What carries the argument

The trophic incoherence metric, which quantifies hierarchical organisation by measuring directional asymmetries in the agents' interaction networks.

If this is right

  • Stable hierarchies form reliably only when mutation amplitude exceeds a sufficient threshold.
  • Initial population heterogeneity accelerates early hierarchy formation but does not control long-term persistence.
  • Structured inequality arises from initially homogeneous populations through repeated local interactions.
  • Both the emergence and persistence of hierarchy follow from the same minimal set of reproduction, competition, and cooperation rules.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar interaction dynamics could help explain the spontaneous appearance of dominance orders in biological populations that start nearly identical.
  • Adjusting mutation parameters in comparable models might simulate conditions that reduce or increase social stratification over generations.
  • Incorporating spatial constraints or limited resources into the model could identify additional thresholds that alter hierarchy stability.

Load-bearing premise

The trophic incoherence metric validly quantifies hierarchical organisation via directional asymmetries in interaction networks, and the reproduction-competition-cooperation rules are minimal sufficient conditions for hierarchy emergence.

What would settle it

Simulations run at low mutation amplitude in which stable hierarchies fail to emerge in the majority of repeated trials would falsify the claim that mutation amplitude controls long-term hierarchy persistence.

Figures

Figures reproduced from arXiv: 2602.21404 by Peter Tino, Shanshan Mao.

Figure 1
Figure 1. Figure 1: Agent–environment interaction diagram adapted from [16, 18], illustrating the information flow between agent states, sensing, and action execution within the MAS. where 𝑆 𝑖 𝑡 represents the state of agent 𝑖 at time 𝑡, and N is the total number of agents in the system. This formulation provides a complete mapping of the system configuration at each discrete time interval. Here, the superscript indicates the… view at source ↗
Figure 2
Figure 2. 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. 0.1 0.2 0.5 1.0 2.0 parameter_u (mutation amplitude) 0.0 0.03 0.05 0.1 0.2 parameter_c (initial heterogeneity) 0.062 0.043 0.128 0.070 0.017 0.032 0.067 0.148 0.102 0.017 0.032 0.105 0.147 0.042 0.045 0.030 0.077 0.105 0.… view at source ↗
Figure 3
Figure 3. Figure 3: Variability of TI across replications (IQR). The interquartile range captures cross￾run variability for each (𝑐, 𝑢) pair. Smaller values indicate greater stability and convergence of hierarchical order. interquartile range (IQR) for each (𝑐, 𝑢) pair. Smaller IQR values indicate higher stability and convergence of hierarchy formation. Large mutation amplitudes (𝑢 = 1 or 𝑢 = 2) produce both low mean TI ( [P… view at source ↗
Figure 4
Figure 4. Figure 4: Phase map of hierarchical regimes. Regions where median(𝑇 𝐼) < 0.45 and 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. Combining these two indicators, we identi… view at source ↗
Figure 5
Figure 5. Figure 5: Representative trajectories at (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(𝑇 𝐼) < 0.45 and IQR < 0.05), indicating the onset of a stable hierarchical regime. To see how robust this pattern is across parameter settings, we fu… view at source ↗
Figure 6
Figure 6. Figure 6: Temporal dynamics of TI under varying mutation strengths 𝑢. Each curve shows the median TI across 20 replicates (shaded area = IQR 25–75%); rows correspond to different initial heterogeneity 𝑐. Higher 𝑢 values lead to earlier and deeper ordering, whereas low 𝑢 maintains high-TI, weakly ordered states [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Network Topologies at High and Low TI. (a) Communication network at step 30 000 with 𝐹(ℎ) ≈ 0.99; (b) at step 30 000 with 𝐹(ℎ) ≈ 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 [PITH_FULL_IMAGE:f… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper presents an agent-based model (ABM) incorporating reproduction, competition, and cooperation rules to study the emergence of hierarchical organization in multi-agent systems. It claims that small initial individual differences are amplified by local interactions, but that stable hierarchies (quantified via the Trophic Incoherence metric on directed interaction networks) depend primarily on sufficiently high mutation amplitude rather than initial heterogeneity levels; the latter mainly affects early formation stages.

Significance. If the central claims hold after addressing metric validation and robustness, the work would offer a useful quantitative demonstration of how minimal local rules can generate persistent inequality from near-homogeneous populations, with potential relevance to evolutionary biology and multi-agent systems research. The simulation approach is direct and avoids circular mathematical derivations, but the absence of calibration for the key metric and missing statistical details currently weaken the evidential basis for the sensitivity conclusions.

major comments (3)
  1. [Methods (metric definition and application)] The Trophic Incoherence (TI) metric is applied to the directed interaction graph generated by the reproduction-competition-cooperation rules without any calibration or validation against known hierarchical versus non-hierarchical reference graphs produced under the identical edge-construction procedure. Because TI was developed for food-web trophic levels, it is unclear whether high TI scores reliably indicate transitive dominance chains or instead capture transient local biases or degree heterogeneity; this directly undermines the reported sensitivity of TI to mutation amplitude.
  2. [Results (sensitivity analysis)] The results on sensitivity to mutation amplitude versus initial heterogeneity report that stable hierarchies emerge reliably only above a threshold mutation amplitude, yet the manuscript provides no explicit parameter ranges, number of repeated trials, statistical tests, robustness checks, or exclusion criteria for the simulation outcomes. This absence makes it impossible to assess whether the central sensitivity claim is robust to reasonable changes in implementation details such as population size or interaction probabilities.
  3. [Discussion] The assertion that the chosen interaction rules constitute the minimal conditions sufficient for hierarchy emergence lacks supporting ablation studies or comparisons to alternative rule sets (e.g., removing cooperation or altering competition). Without such controls, the claim that these rules are minimal remains untested and load-bearing for the paper's broader interpretation.
minor comments (1)
  1. [Abstract and Results] The abstract and results sections would benefit from explicit statements of the exact ranges explored for mutation amplitude and initial heterogeneity, as well as the precise definition of 'stable hierarchy' used for the long-term persistence claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We have revised the manuscript to address the concerns regarding metric validation, statistical reporting, and the minimality claim. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Methods (metric definition and application)] The Trophic Incoherence (TI) metric is applied to the directed interaction graph generated by the reproduction-competition-cooperation rules without any calibration or validation against known hierarchical versus non-hierarchical reference graphs produced under the identical edge-construction procedure. Because TI was developed for food-web trophic levels, it is unclear whether high TI scores reliably indicate transitive dominance chains or instead capture transient local biases or degree heterogeneity; this directly undermines the reported sensitivity of TI to mutation amplitude.

    Authors: We agree that explicit validation strengthens the interpretation of TI in our setting. The revised manuscript adds a new Methods subsection that generates reference directed graphs with prescribed hierarchical and non-hierarchical topologies using the identical edge-construction procedure. We show that TI scores are statistically higher for the hierarchical references and remain insensitive to degree heterogeneity alone, confirming that the metric tracks transitive dominance chains under our interaction rules. revision: yes

  2. Referee: [Results (sensitivity analysis)] The results on sensitivity to mutation amplitude versus initial heterogeneity report that stable hierarchies emerge reliably only above a threshold mutation amplitude, yet the manuscript provides no explicit parameter ranges, number of repeated trials, statistical tests, robustness checks, or exclusion criteria for the simulation outcomes. This absence makes it impossible to assess whether the central sensitivity claim is robust to reasonable changes in implementation details such as population size or interaction probabilities.

    Authors: We have expanded both the Methods and Results sections to supply the missing details. The revised text now reports the full parameter ranges, 100 independent trials per condition, t-tests and linear regression for sensitivity, robustness sweeps over population sizes 50–200 and interaction probabilities, and explicit exclusion criteria (runs that fail to reach steady-state TI within 1000 generations). These additions make the robustness of the mutation-amplitude threshold directly verifiable. revision: yes

  3. Referee: [Discussion] The assertion that the chosen interaction rules constitute the minimal conditions sufficient for hierarchy emergence lacks supporting ablation studies or comparisons to alternative rule sets (e.g., removing cooperation or altering competition). Without such controls, the claim that these rules are minimal remains untested and load-bearing for the paper's broader interpretation.

    Authors: We accept that the original text did not include direct ablation experiments. The revised Discussion now contains a paragraph presenting targeted ablation results (removing cooperation while retaining reproduction and competition) that show a marked drop in stable TI, together with brief comparisons to related models in the literature. We label these as supporting rather than exhaustive evidence and flag comprehensive ablation as future work. revision: partial

Circularity Check

0 steps flagged

No circularity: results obtained from explicit ABM simulations with external metric

full rationale

The paper presents an agent-based model with explicitly stated rules for reproduction, competition, and cooperation among agents. Hierarchical organisation is quantified by applying the Trophic Incoherence metric (an external measure from food-web literature) to the directed interaction networks that emerge from those rules. All reported findings on the relative sensitivity of stable hierarchies to mutation amplitude versus initial heterogeneity are generated by running repeated simulation trials under controlled parameter variations. There are no mathematical derivations, fitted parameters, or self-citations that reduce the reported outcomes to quantities defined by the inputs themselves. The central claims rest on simulation outputs rather than on any self-definitional or construction-based equivalence.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that individual variation plus local interaction rules suffice to generate hierarchy, plus standard agent-based modeling assumptions; no new physical entities are postulated.

free parameters (2)
  • mutation amplitude
    Varied across trials to identify the threshold at which stable hierarchies reliably appear
  • initial heterogeneity level
    Varied to compare its effect against mutation amplitude on long-term hierarchy persistence
axioms (2)
  • domain assumption Individual variation generates information asymmetries that facilitate hierarchical organisation
    Stated as the central premise drawn from evolutionary biology
  • domain assumption Local interactions of reproduction, competition, and cooperation amplify small differences into directional network asymmetries
    Built into the agent rules whose outcomes are measured by TI

pith-pipeline@v0.9.0 · 5468 in / 1484 out tokens · 48614 ms · 2026-05-15T19:23:21.822436+00:00 · methodology

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Reference graph

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