Bounded Criticality in Multi-Agent Systems

We investigate cascade dynamics in interacting AI agent systems under increasing external load.

Our results suggest that these systems exhibit bounded criticality, characterized by a rapid but finite transition in collapse probability, rather than classical divergence.

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Key Findings

• A well-defined critical threshold:

  σ_c ≈ 0.48

• A smooth transition in collapse probability across load levels

• A finite peak in susceptibility (χ = dP/dk), indicating:

  → non-divergent critical behavior

• Empirical scaling consistent with:

  z ≈ 1.39

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Main Result [collapse]

Figure 1 — Collapse transition across network topologies.
Collapse probability as a function of external load k for multiple interaction structures (Ring, Erdős–Rényi, Small-world, Scale-free).

All configurations exhibit a smooth transition without divergence.
The vertical dashed line marks the estimated critical threshold σ_c ≈ 0.48.

This behavior is consistent with a bounded critical regime, where instability emerges but remains finite.

Figure 2 — Finite-size signatures of bounded criticality in multi-agent systems. (A) Collapse probability as a function of external load $k$ for different system sizes $L$. The transition exhibits a smooth but size-dependent shift, indicating a finite-size pseudo-critical point. (B) Susceptibility proxy $\chi = dP/dk$ displays pronounced peaks for all system sizes, confirming the presence of critical-like fluctuations. (C) Peak susceptibility $\chi_{\max}(L)$ as a function of system size. While $\chi_{\max}$ increases with $L$, the growth does not exhibit clear divergence, suggesting a bounded critical regime rather than classical second-order phase transition behavior.

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B. Susceptibility Peaks and Critical-Like Fluctuations

To quantify the transition, we compute a susceptibility proxy defined as:

$$ \chi(k) = \frac{dP}{dk} $$

Figure 2B shows $\chi(k)$ for different system sizes.

For all values of $L$, the susceptibility exhibits a pronounced peak at a characteristic load $k_c(L)$, which we identify as the pseudo-critical point of the system. The presence of these peaks indicates enhanced fluctuations near the transition, a hallmark of critical-like behavior.

Notably, the peak height increases with system size, suggesting the amplification of fluctuations as the system grows.

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C. Finite-Size Scaling and Bounded Criticality

To investigate scaling behavior, we extract the peak susceptibility $\chi_{\max}(L)$ for each system size and analyze its dependence on $L$ (Figure 2C).

In classical second-order phase transitions, one expects:

$$ \chi_{\max}(L) \sim L^{\gamma/\nu} $$

indicating divergence in the thermodynamic limit.

However, in our system, while $\chi_{\max}$ increases with $L$, the growth is sublinear and does not exhibit clear divergence. This suggests that the system does not belong to a conventional universality class with divergent susceptibility.

Instead, the observed behavior is consistent with a bounded critical regime, in which:

• critical-like fluctuations emerge,
• susceptibility peaks are present,
• but no divergence occurs in the large-$L$ limit.

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D. Interpretation

The coexistence of:

• smooth collapse transitions,
• well-defined susceptibility peaks,
• and non-diverging $\chi_{\max}$,

indicates that the system operates in a regime distinct from classical critical phenomena.

We interpret this as evidence for bounded criticality, a dynamical regime in which systems approach instability but remain constrained by intrinsic regulatory mechanisms (e.g., nonlinear feedback or memory effects).

Such behavior is particularly relevant for multi-agent AI systems, where cascading failures can emerge without leading to unbounded systemic collapse.

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Analysis Pipeline

This repository provides a minimal, reproducible pipeline:

Simulation → Data → Analysis → Logs → Figures

Components

• simulation.py — generates collapse probability curves
• analysis.py — computes susceptibility and detects σ_c
• data/ — input dataset (collapse probability vs load)
• results/log/ — derived metrics and transition summaries
• results/figures/ — publication-style visualizations

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Reproducibility

To reproduce results locally:

pip install -r requirements.txt

python src/simulation.py
python src/analysis.py

Outputs:

data/collapse_data.csv
results/log/ (σ_c, χ, metadata)
results/figures/ (final plots)
Definition of Susceptibility

We define a susceptibility proxy:

χ = dP/dk

where:

P = collapse probability
k = external load

The peak of χ(k) determines the critical threshold σ_c.

Interpretation

Unlike classical phase transitions:

No divergence in χ is observed
The transition remains smooth and bounded
The system exhibits rapid but finite instability growth

This suggests a bounded criticality regime, distinct from traditional absorbing-state transitions.

Limitations

This repository provides a minimal and abstracted model of cascade dynamics.

The underlying micro-level update rules are simplified
Topology-dependent mechanisms are not fully parameterized
Finite-size scaling (FSS) is not yet included

The implementation is intended as a reproducible demonstration, not a complete physical theory.

Notes on Scope
This repository focuses on observable behavior (collapse curves, χ, σ_c)
Detailed micro-dynamics and parameter optimization are not included in this public version
Repository Structure
bounded-criticality/
│
├── data/
├── src/
├── results/
│   ├── figures/
│   └── log/
├── README.md
└── requirements.txt
Status

This is an early-stage research release intended for:

reproducibility
discussion
preliminary validation

Further extensions will include:

topology-dependent simulations
finite-size scaling analysis
extended dynamical modeling
License

MIT License (or specify your preferred license)