Emergent Macro-Criticality from Micro-Critical Agents

1. Problem Statement

The paper addresses a fundamental gap in the understanding of criticality in complex systems: how do critical dynamics propagate from the microscopic level (individual agents) to the macroscopic level (collective behavior)?

While criticality is widely proposed as a principle for optimal information processing in biological and artificial systems (e.g., neural avalanches), it remains unclear whether tuning individual components to be near-critical is sufficient to generate collective critical-like behavior. The authors investigate whether internal critical dynamics alone can drive a multi-agent system to a critical state, or if additional constraints regarding interaction topology and connectivity are required.

2. Methodology

The authors employ a multi-agent simulation framework that separates dynamics into two distinct levels:

Micro-Level: The Internal Reservoir

• Structure: Each agent contains a discrete-time stochastic recurrent network (a reservoir) of $N_{neurons} = 64$ binary neurons.
• Dynamics: The reservoir models neuronal avalanche dynamics using a branching process. Neuron $i$ activates neuron $j$ with a transmission probability $W_{i,j}$.
• Control Parameter: The system uses a homogeneous transmission probability $p_{micro}$ for all connections (where $i \neq j$).
• Critical Point: The theoretical critical point for an isolated reservoir is $p_{micro}^c = \frac{1}{N_{neurons}-1}$. The authors tune $p_{micro}$ around this value to create subcritical, near-critical, and supercritical regimes.

Macro-Level: Spatial Interaction

• Environment: 256 agents are placed in a 2D toroidal space.
• Sensing & Signaling: Agents emit binary light signals. They sense neighbors within a fixed vision radius ($v_r$).
• Coupling: Sensory input is discretized into $E=8$ exteroceptive sectors. These inputs perturb the internal reservoir of the agent. The reservoir's output determines the agent's next light state.
• Simulation Protocol:
  • Simulations start with a localized perturbation (one agent ON).
  • The system evolves for 1000 macro-timesteps.
  • Avalanche Definition: An avalanche is defined as a contiguous period of non-zero population-wide light activity, bounded by silence ($A_l(t)=0$).
  • Parameters Swept: The study systematically varies $p_{micro}$ (internal dynamics) and $v_r$ (interaction topology).

Analysis Metrics

• Avalanche Statistics: Size ($S$) and Duration ($T$) distributions are analyzed.
• Power-Law Fitting: The authors use the Clauset et al. (2009) framework to fit discrete power laws ($P(x) \propto x^{-\alpha}$) to the tails of the distributions.
• Criticality Score ($L$): A heuristic metric quantifying the deviation from theoretical critical exponents:
  $$L = |\alpha_S - 1.5| + |\alpha_T - 2| + KS_S + KS_T$$
  Lower $L$ indicates statistics closer to the mean-field critical branching prediction ($\alpha_S=1.5, \alpha_T=2$).
• Effective Branching Ratio ($\sigma_{eff}$): A measure of global activity flux to identify regime transitions.

3. Key Contributions

1. Decoupling Micro and Macro Criticality: The paper demonstrates that near-critical internal dynamics are not sufficient to produce collective criticality. The macro-level critical regime is displaced relative to the micro-level critical point.
2. Role of Topology: The emergence of scale-free behavior is shown to depend critically on the effective connectivity of the interaction network (governed by $v_r$), rather than just the internal agent parameters.
3. Subcritical Robustness: A novel finding is that slightly subcritical microscopic regimes ($p_{micro} < p_{micro}^c$) actually support a wider range of macroscopic near-critical behavior compared to regimes tuned exactly to the micro-critical point.
4. Quantitative Shift: The authors map the "phase diagram" of the system, showing how the critical regime shifts systematically as the vision radius (and thus network connectivity) changes.

4. Key Results

The Criticality Landscape

• Phase Diagram: The heatmap of the criticality score $L(p_{micro}, v_r)$ reveals three distinct regimes:
  1. Silence: Low $p_{micro}$ and low $v_r$ (activity dies out).
  2. Saturation: High $p_{micro}$ and high $v_r$ (persistent, dense activity with no distinct avalanches).
  3. Near-Critical Band: A narrow, curved band of high criticality scores between the extremes.
• Displacement: The band of macro-criticality does not align with the theoretical micro-critical point ($p_{micro}^c$). Instead, it shifts to lower $p_{micro}$ values as $v_r$ increases.

Connectivity and Propagation

• Percolation Threshold: The system undergoes a topological transition at $v_r \approx 47$, where a "giant connected component" emerges. Below this, the network is fragmented; above it, global propagation is possible.
• Effective Branching: As $v_r$ increases, the transition of the effective branching ratio ($\sigma_{eff}$) toward 1 (indicating sustained activity) occurs at progressively lower values of $p_{micro}$. This confirms that spatial coupling enhances activity propagation, compensating for lower internal transmission probabilities.

Avalanche Statistics

• Exponent Deviation: While the system exhibits scale-free (heavy-tailed) avalanches, the fitted exponents deviate from mean-field predictions.
  • Observed $\alpha_S \approx 1.7$ (vs. theoretical 1.5).
  • Observed $\alpha_T \approx 1.9 - 2.0$.
• Interpretation: These deviations are attributed to finite-size effects and spatial constraints (non-homogeneous mixing), which violate the assumptions of simple branching processes.

Robustness of Subcritical Micro-Regimes

• Integrating the criticality score over connectivity values reveals that operating the internal reservoirs slightly below the critical point ($p_{micro} < p_{micro}^c$) yields the most robust collective near-critical behavior across a wide range of interaction topologies.

5. Significance and Implications

• Evolutionary Insight: The findings suggest that biological systems embedded in large interaction networks (e.g., multicellular organisms, neural colonies) may not need to tune individual components to exact criticality. Instead, they may operate in a slightly subcritical regime to ensure robust collective criticality despite variations in network structure.
• Design Principles for AI/Robotics: For multi-agent systems and reservoir computing, optimizing collective intelligence requires tuning the interaction structure (connectivity) in tandem with internal dynamics, rather than optimizing agents in isolation.
• Theoretical Clarification: The work clarifies that "criticality" is a cross-scale property. It is an emergent phenomenon resulting from the interplay between internal dynamics and the constraints of the interaction network, challenging the view that criticality is solely an intrinsic property of individual units.

Conclusion

The paper concludes that collective near-critical behavior is not an intrinsic consequence of tuning individual agents to criticality. It is an emergent property governed by the interplay between internal dynamics and the topology of activity propagation. Specifically, a slight deviation from micro-criticality (subcriticality) combined with appropriate macroscopic connectivity creates the most robust conditions for scale-free collective dynamics.