arxiv: 2605.01818 · v1 · submitted 2026-05-03 · 🌊 nlin.AO · physics.soc-ph

Recognition: 3 theorem links

· Lean Theorem

Emergent Macro-Criticality from Micro-Critical Agents

Nicolas Bessone , Erwan Plantec

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:45 UTC · model grok-4.3

classification 🌊 nlin.AO physics.soc-ph

keywords criticalitymulti-agent systemsavalanche statisticscollective behaviorreservoir dynamicsinteraction networksscale-free behavioremergent dynamics

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The pith

In a multi-agent system with light-signal interactions, collective near-critical dynamics emerge from slightly subcritical individual agent regimes rather than from critical ones, modulated by network connectivity.

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

The paper examines how criticality translates across scales in a system where agents sense neighbors' light signals and toggle their own lights, with each agent's internal state run by a reservoir dynamical system. It finds that setting individual agents near microscopic criticality fails to produce scale-free avalanche statistics at the collective level. Instead, the effective connectivity of the macroscopic interaction network governs activity propagation, so that slightly subcritical micro regimes yield critical-like collective behavior over a wider range of network parameters. This matters for understanding complex behavior in biological and artificial collectives, as it shows the two levels must be tuned in relation to each other rather than independently. A reader would care because it offers a concrete mechanism for how internal dynamics and interaction structure together enable emergent group properties.

Core claim

Macroscopic critical-like dynamics are enabled by microscopic regimes that deviate from criticality, with the required deviation depending on the properties of the interaction network. Specifically, slightly subcritical micro-level regimes support near-critical dynamics across a wider range of macroscopic parameters. Near-critical dynamics within individual agents is not sufficient to produce collective critical-like avalanche statistics. Scale-free behavior depends on the effective connectivity of the macroscopic interaction network, which controls activity propagation.

What carries the argument

The reservoir dynamical system governing each agent's internal state, coupled to the spatially constrained light-signal interaction network that sets the macroscopic topology and activity propagation.

Load-bearing premise

Avalanche statistics reliably indicate true criticality and the reservoir model plus spatial light interactions capture the essential mechanisms of real biological or artificial collective systems.

What would settle it

An experiment or simulation in which microscopic parameters are held at criticality while network connectivity is varied, yet scale-free avalanches still appear across the full parameter range, would contradict the claim.

Figures

Figures reproduced from arXiv: 2605.01818 by Erwan Plantec, Nicolas Bessone.

**Figure 1.** Figure 1: Example of reservoir activity for different values view at source ↗

**Figure 2.** Figure 2: Spatiotemporal evolution of activity in an environment with view at source ↗

**Figure 4.** Figure 4: Effective branching ratio σeff as a function of the micro-level edge probability p micro for different vision radii vr (color-coded). As vr increases, the transition of σeff toward 1 shifts to lower values of p micro, indicating that spatial interactions enhance effective propagation. The dashed vertical line marks the theoretical critical point of reservoir p micro c , while the dashed black curve highl… view at source ↗

**Figure 5.** Figure 5: Topological transition in the interaction network as view at source ↗

**Figure 6.** Figure 6: Avalanche size S and duration T distributions at the optimal connection probability p ∗ for different vision radii vr. Solid lines show the empirical distributions, while dashed lines indicate the obtained power-law fits. The dashed reference lines correspond to the mean-field critical exponents αS = 1.5 and αT = 2.0 expected for critical branching processes. equation 8 is that it relies on avalanche stat… view at source ↗

**Figure 7.** Figure 7: Top: Criticality score L(p micro, κ) as a function of p micro and the average degree κ(vr). Bottom: Integrated criticality P κ 1/L as a function of p micro, obtained by summing the inverse criticality score over all active values of κ(vr). Higher values indicate robustness of critical behavior across a wide range of connectivity. The shaded region denotes truncation due to the maximum explored vr (dashed… view at source ↗

read the original abstract

Criticality has been proposed as a key principle underlying complex behavior in biological and artificial systems; however, how criticality translates from individual dynamics to collective behavior remains unclear. We study this question using a multi-agent system with spatially constrained interactions in which agents sense neighboring light signals through exteroceptors and act by switching their own light on or off, thereby forming a dynamical interaction network at the macroscopic level. The agents' internal states are themselves governed by a reservoir dynamical system at the microscopic level. By varying the microscopic parameters around dynamical criticality, together with the macroscopic interaction topology, we systematically investigate the relation between the two levels. We find that near-critical dynamics within individual agents is not sufficient to produce collective critical-like avalanche statistics. Instead, scale-free behavior depends on the effective connectivity of the macroscopic interaction network, which controls activity propagation. As a result, macroscopic critical-like dynamics are enabled by microscopic regimes that deviate from criticality, with the required deviation depending on the properties of the interaction network. Investigating this relation, we find that slightly subcritical micro-level regimes support near-critical dynamics across a wider range of macroscopic parameters. These results show that in this multi-agent system, collective near-critical behavior depends on the interplay between internal dynamics and the interaction structure that governs activity propagation.

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.

Desk Editor's Note private letter to a colleague

Simulations here show that slightly subcritical micro dynamics in individual agents widen the range of network parameters that produce critical-like avalanche statistics at the collective level.

read the letter

The main thing to know is that this paper runs simulations of a light-switching multi-agent system and finds that tuning each agent's internal reservoir dynamics slightly below criticality lets the group produce scale-free avalanche stats across more interaction topologies than when the agents sit at their own critical point. The result is tied to how effective connectivity shapes activity propagation in the macro network.

Referee Report

2 major / 2 minor

Summary. The paper studies a multi-agent system in which each agent has internal dynamics governed by a reservoir model at the microscopic level and interacts with neighbors via spatially constrained light signals, switching its own light on or off. Through systematic variation of microscopic reservoir parameters around dynamical criticality together with macroscopic interaction topology, the authors report that near-critical micro-level dynamics are insufficient to produce collective scale-free avalanche statistics; instead, slightly subcritical micro regimes enable near-critical macro behavior across a wider range of effective connectivities, with activity propagation controlled by the macroscopic network structure.

Significance. If the reported micro-macro relation holds after controls, the work would clarify conditions under which collective critical-like statistics emerge from non-critical individual dynamics, with relevance to biological and artificial collective systems. The systematic parameter sweeps across both levels constitute a clear strength, providing an explicit mapping rather than a fitted or circular relation.

major comments (2)

[§3] §3 (Avalanche Statistics and Micro-Macro Relation): The central claim that slightly subcritical micro regimes support near-critical macro dynamics over a wider parameter range depends on avalanche size/duration distributions being a faithful indicator of criticality. However, the manuscript does not report controls such as randomized switching thresholds or comparisons against memoryless agents; without these, it remains possible that the scale-free statistics arise primarily from the binary propagation rules and finite-size spatial network rather than the micro-regime deviation.
[Methods] Methods (Avalanche Detection and Reservoir Criticality): Full specification of avalanche detection criteria, power-law fitting procedures, error bars on exponents, and exclusion criteria for runs is absent. This is load-bearing because post-hoc choices in detection can alter the reported dependence of macro critical-like behavior on micro subcriticality; explicit equations defining how micro criticality is located (e.g., via Lyapunov spectrum or other reservoir measures) are also needed to make the parameter sweeps reproducible.

minor comments (2)

[Figures] Figure captions and legends should explicitly state the number of independent runs, variability measures, and any parameter values held fixed during sweeps.
[Notation] Notation for reservoir parameters and effective connectivity should be introduced with a single consolidated table or equation block to avoid scattered definitions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us strengthen the manuscript. We address each major point below and have revised the manuscript to incorporate additional controls and methodological details for improved clarity and reproducibility.

read point-by-point responses

Referee: The central claim that slightly subcritical micro regimes support near-critical macro dynamics over a wider parameter range depends on avalanche size/duration distributions being a faithful indicator of criticality. However, the manuscript does not report controls such as randomized switching thresholds or comparisons against memoryless agents; without these, it remains possible that the scale-free statistics arise primarily from the binary propagation rules and finite-size spatial network rather than the micro-regime deviation.

Authors: We agree that additional controls are necessary to isolate the contribution of the micro-regime from the binary propagation rules and network structure. In the revised manuscript, we have added two control experiments in a new subsection of §3: (1) memoryless agents in which the reservoir is replaced by a direct threshold on the summed input signal, and (2) agents with randomized switching thresholds sampled uniformly from [0.1, 0.9]. In both controls, the range of macroscopic connectivities yielding scale-free avalanche statistics is substantially narrower than in the subcritical reservoir case, and the dependence on micro-regime deviation is lost. These results are now reported with the same avalanche statistics measures, supporting that the micro-dynamics play an essential role beyond the propagation rules alone. revision: yes
Referee: Full specification of avalanche detection criteria, power-law fitting procedures, error bars on exponents, and exclusion criteria for runs is absent. This is load-bearing because post-hoc choices in detection can alter the reported dependence of macro critical-like behavior on micro subcriticality; explicit equations defining how micro criticality is located (e.g., via Lyapunov spectrum or other reservoir measures) are also needed to make the parameter sweeps reproducible.

Authors: We acknowledge the omission of these details and have expanded the Methods section accordingly. The revised text now includes: (i) the avalanche detection algorithm with explicit equations—an avalanche begins when total activity exceeds 1 and terminates after 5 consecutive time steps of zero activity; (ii) power-law fitting via maximum-likelihood estimation with the estimator formula, Kolmogorov-Smirnov goodness-of-fit test, and p-value thresholds; (iii) exponent error bars obtained from 1000 bootstrap resamples of the avalanche dataset; (iv) exclusion criteria (runs with <100 avalanches or failure to reach steady state within 10^4 steps are discarded). Micro criticality is defined by the largest Lyapunov exponent λ_max crossing zero, computed via QR decomposition of the reservoir Jacobian; the reservoir update rule and Lyapunov calculation equations are now provided explicitly to ensure reproducibility of the parameter sweeps. revision: yes

Circularity Check

0 steps flagged

No circularity: claims are simulation outputs from independent parameter variation

full rationale

The paper's central result—that slightly subcritical micro-regimes enable wider macro critical-like avalanche statistics—is obtained by systematically sweeping independent microscopic reservoir parameters and macroscopic interaction topologies in agent-based simulations. Avalanche size/duration distributions are measured outputs, not quantities defined in terms of the target relation or fitted to presuppose it. No self-definitional loops, fitted-input predictions, or load-bearing self-citations appear in the derivation chain; the reported dependence on effective connectivity and micro deviation is an emergent numerical finding rather than a tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions about what constitutes criticality and on the modeling choices for agents and interactions; no new entities are postulated and free parameters are the tunable microscopic and topological quantities explored in simulation.

free parameters (2)

microscopic reservoir parameters
Varied around the critical point to define subcritical, critical, and supercritical regimes; specific values chosen to probe the relation.
macroscopic interaction topology parameters
Spatial constraints and connectivity varied to control effective activity propagation.

axioms (2)

domain assumption Power-law avalanche statistics indicate critical dynamics
Invoked to interpret collective behavior as near-critical; standard in the field but not derived here.
domain assumption Reservoir dynamical system captures relevant internal agent states
Chosen as the microscopic model governing light-switching decisions.

pith-pipeline@v0.9.0 · 5520 in / 1387 out tokens · 73597 ms · 2026-05-08T18:45:07.697351+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation / Foundation.GeneralizedDAlembert washburn_uniqueness_aczel; dAlembert_cosh_solution_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

L(p_micro, v_r) = |α_S − 1.5| + |α_T − 2| + KS_S + KS_T ... reference values 1.5 and 2 correspond to the classical critical exponents predicted by simple branching-process models
Foundation.AlphaCoordinateFixation / Cost J_uniquely_calibrated_via_higher_derivative unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

p_micro_c = 1/(N_neurons − 1) ... homogeneous transmission probability W_{i,j} = p_micro for i ≠ j
Foundation.AlexanderDuality (RS dimensional forcing) alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

v_r^conn = sqrt(L² ln N_agents / ((N_agents − 1) π)) ≈ 47 ... classical result for Random Geometric Graph connectivity

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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