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"""
Self-Organized Criticality (SOC) Analysis for Prey-Predator Cellular Automaton
This module analyzes whether the prey-predator CA exhibits SOC properties:
1. Slow drive: Gradual external parameter changes
2. Build-up of stress with thresholds (storing potential energy)
3. Intermittent release of stress (avalanches/cascades)
4. Self-organization (robustness across parameter variations, not carefully tuned)
We focus on perturbations from initial configurations (not at critical point) and
sample across different configurations with varied parameters beyond just death/birth
rates (e.g., grid size, densities, neighborhood, synchronicity).
"""
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
from matplotlib.gridspec import GridSpec
from typing import List, Tuple, Dict, Optional
import sys
from pathlib import Path
# Add parent directory to path
sys.path.insert(0, str(Path(__file__).parent.parent))
# Import directly from models to avoid __init__ issues
from models.CA import PP
# ============================================================================
# 1. STRESS METRIC & PERTURBATION DYNAMICS
# ============================================================================
def compute_grid_stress(grid: np.ndarray) -> float:
"""
Compute a 'stress' metric for the grid as a proxy for potential energy.
High predator-prey interface regions represent tension/stress.
Stress = normalized count of (predator,empty) and (prey,empty) adjacent pairs.
This represents the gradient/friction that can cause avalanche-like events.
Args:
grid: 2D array with 0=empty, 1=prey, 2=predator
Returns:
Normalized stress value [0, 1]
"""
rows, cols = grid.shape
stress = 0
# Count interfaces (predator or prey adjacent to empty)
for i in range(rows):
for j in range(cols):
cell = grid[i, j]
if cell == 0: # empty cell
# Check all 8 neighbors
for di in [-1, 0, 1]:
for dj in [-1, 0, 1]:
if di == 0 and dj == 0:
continue
ni, nj = (i + di) % rows, (j + dj) % cols
if grid[ni, nj] > 0: # neighbor is prey or predator
stress += 1
# Normalize by maximum possible interfaces
max_stress = rows * cols * 8 # each cell can have 8 neighbors
return stress / max_stress if max_stress > 0 else 0.0
def compute_population_variance(grids_history: List[np.ndarray]) -> Tuple[np.ndarray, np.ndarray]:
"""
Compute prey and predator population variance over time.
High variance indicates intermittent release events.
Args:
grids_history: List of grid snapshots over time
Returns:
Tuple: (prey_variance_rolling, predator_variance_rolling)
"""
prey_pops = np.array([(g == 1).sum() for g in grids_history])
pred_pops = np.array([(g == 2).sum() for g in grids_history])
window = max(5, len(grids_history) // 10) # rolling window
prey_changes = np.abs(np.diff(prey_pops))
pred_changes = np.abs(np.diff(pred_pops))
# Pad at the beginning to match original length
prey_var = np.concatenate([[prey_changes[0]] * (window - 1),
np.convolve(prey_changes, np.ones(window)/window, mode='valid')])
pred_var = np.concatenate([[pred_changes[0]] * (window - 1),
np.convolve(pred_changes, np.ones(window)/window, mode='valid')])
# Ensure exact length match
prey_var = prey_var[:len(grids_history)]
pred_var = pred_var[:len(grids_history)]
return prey_var, pred_var
def detect_avalanche_events(grids_history: List[np.ndarray],
population_change_threshold: float = 0.1) -> List[Tuple[int, float]]:
"""
Detect avalanche events as rapid changes in total population.
Args:
grids_history: List of grid snapshots
population_change_threshold: Fraction of grid change to trigger detection
Returns:
List of (time_step, magnitude) tuples
"""
total_pops = np.array([(g > 0).sum() for g in grids_history])
max_pop = total_pops.max()
if max_pop == 0:
return []
changes = np.abs(np.diff(total_pops))
threshold = population_change_threshold * max_pop
avalanches = []
in_event = False
event_start = 0
event_magnitude = 0
for i, change in enumerate(changes):
if change > threshold:
if not in_event:
event_start = i
in_event = True
event_magnitude = max(event_magnitude, change)
else:
if in_event:
avalanches.append((event_start, event_magnitude / max_pop))
in_event = False
event_magnitude = 0
return avalanches
# ============================================================================
# 2. PARAMETER SAMPLING WITH VARIED CONFIGURATIONS
# ============================================================================
def sample_parameter_configurations(n_samples: int = 10,
base_seed: int = 42) -> List[Dict]:
"""
Generate diverse parameter configurations.
Varies: grid size, initial densities, rates, neighborhood, synchronicity.
Args:
n_samples: Number of configurations to generate
base_seed: Base random seed
Returns:
List of configuration dicts
"""
configs = []
rng = np.random.RandomState(base_seed)
for i in range(n_samples):
# Vary grid size (smaller = more fluctuations, larger = more stable?)
grid_size = rng.choice([16, 32, 48, 64])
# Vary initial densities (more heterogeneous = more stress buildup?)
prey_density = rng.uniform(0.1, 0.4)
pred_density = rng.uniform(0.02, 0.15)
# Vary parameters beyond just death/birth rates
config = {
"seed": base_seed + i,
"rows": grid_size,
"cols": grid_size,
"densities": (prey_density, pred_density),
"neighborhood": rng.choice(["neumann", "moore"]),
"synchronous": rng.choice([True, False]),
# Vary rate parameters
"prey_death": rng.uniform(0.01, 0.10),
"predator_death": rng.uniform(0.05, 0.20),
"prey_birth": rng.uniform(0.10, 0.35),
"predator_birth": rng.uniform(0.10, 0.30),
}
configs.append(config)
return configs
# ============================================================================
# 3. SLOW DRIVE & STRESS BUILDUP WITH PERTURBATIONS
# ============================================================================
def run_soc_perturbation_experiment(config: Dict,
n_equilibration: int = 100,
n_observation: int = 200,
perturbation_step: int = 50) -> Dict:
"""
Run a single SOC experiment with slow parameter drift (drive) and
perturbations from non-critical initial conditions.
The experiment:
1. Initialize CA with given config (not at "critical point")
2. Run equilibration steps (slow drive builds up stress)
3. Perturb one parameter gradually
4. Observe stress buildup and release events
Args:
config: Configuration dict from sample_parameter_configurations()
n_equilibration: Steps before perturbation (building stress)
n_observation: Steps during/after perturbation (observing avalanches)
perturbation_step: Which step to start perturbation
Returns:
Dict with results: stress_history, populations, avalanches, etc.
"""
# Create PP automaton
ca = PP(
rows=int(config["rows"]),
cols=int(config["cols"]),
densities=tuple(float(d) for d in config["densities"]),
neighborhood=config["neighborhood"],
params={
"prey_death": float(config["prey_death"]),
"predator_death": float(config["predator_death"]),
"prey_birth": float(config["prey_birth"]),
"predator_birth": float(config["predator_birth"]),
},
seed=int(config["seed"]),
synchronous=False, # Use async mode since sync is not fully implemented
)
# Run equilibration: slow drive allows stress to build
stress_history = []
grids_history = []
prey_pops = []
pred_pops = []
param_history = [] # track parameter drift
total_steps = n_equilibration + n_observation
for step in range(total_steps):
# Slow parameter drift (drive): gradually increase predator death
# This is the "slow drive" that accumulates stress without immediate release
if step >= perturbation_step:
progress = (step - perturbation_step) / (total_steps - perturbation_step)
drift_amount = 0.05 * progress # drift up to +0.05
ca.params["predator_death"] = config["predator_death"] + drift_amount
# Record state before update
stress = compute_grid_stress(ca.grid)
stress_history.append(stress)
grids_history.append(ca.grid.copy())
prey_pops.append((ca.grid == 1).sum())
pred_pops.append((ca.grid == 2).sum())
param_history.append(float(ca.params["predator_death"]))
# Update CA
ca.update()
# Detect avalanche events
avalanches = detect_avalanche_events(grids_history, population_change_threshold=0.05)
# Compute variance (intermittent release signature)
prey_var, pred_var = compute_population_variance(grids_history)
# Ensure exact length match with steps (fix any off-by-one errors)
if len(prey_var) < len(grids_history):
prey_var = np.pad(prey_var, (0, len(grids_history) - len(prey_var)), mode='edge')
if len(pred_var) < len(grids_history):
pred_var = np.pad(pred_var, (0, len(grids_history) - len(pred_var)), mode='edge')
results = {
"config": config,
"stress_history": np.array(stress_history),
"prey_populations": np.array(prey_pops),
"pred_populations": np.array(pred_pops),
"param_history": np.array(param_history),
"avalanches": avalanches,
"prey_variance": prey_var,
"pred_variance": pred_var,
"grids_history": grids_history,
"total_steps": total_steps,
"n_equilibration": n_equilibration,
}
return results
# ============================================================================
# 4. ROBUSTNESS ANALYSIS (Criticality across parameters)
# ============================================================================
def analyze_soc_robustness(experiment_results: List[Dict]) -> Dict:
"""
Analyze robustness of critical behavior across diverse parameter configs.
SOC robustness signature: avalanche statistics (frequency, magnitude)
remain relatively consistent across diverse parameter combinations,
indicating self-organization independent of tuning.
Args:
experiment_results: List of results from run_soc_perturbation_experiment()
Returns:
Dict with robustness metrics
"""
avalanche_counts = []
avalanche_magnitudes = []
stress_levels = []
population_variances = []
for result in experiment_results:
if result["avalanches"]:
avalanche_counts.append(len(result["avalanches"]))
mags = [mag for _, mag in result["avalanches"]]
avalanche_magnitudes.extend(mags)
else:
avalanche_counts.append(0)
stress_levels.extend(result["stress_history"].tolist())
population_variances.append(result["prey_variance"].mean())
robustness_metrics = {
"avg_avalanche_count": np.mean(avalanche_counts) if avalanche_counts else 0,
"std_avalanche_count": np.std(avalanche_counts) if avalanche_counts else 0,
"avalanche_magnitude_mean": np.mean(avalanche_magnitudes) if avalanche_magnitudes else 0,
"avalanche_magnitude_std": np.std(avalanche_magnitudes) if avalanche_magnitudes else 0,
"avg_stress": np.mean(stress_levels),
"std_stress": np.std(stress_levels),
"avg_population_variance": np.mean(population_variances),
"coefficient_of_variation_avalanche": (
np.std(avalanche_counts) / np.mean(avalanche_counts)
if np.mean(avalanche_counts) > 0 else np.inf
),
}
return robustness_metrics
# ============================================================================
# 5. VISUALIZATION
# ============================================================================
def visualize_soc_properties(experiment_results: List[Dict],
robustness_metrics: Dict,
output_file: Optional[str] = None):
"""
Visualization of the 4 core SOC properties in prey-predator CA.
Shows:
1. Slow drive: Gradual parameter drift
2. Build-up of stress: Stress accumulation with thresholds
3. Intermittent release: Avalanche cascades and population dynamics
4. Self-organization: Robustness across diverse configurations
Args:
experiment_results: List of experiment results
robustness_metrics: Robustness analysis output
output_file: Optional file path to save figure
"""
fig = plt.figure(figsize=(14, 10))
gs = GridSpec(2, 2, figure=fig, hspace=0.3, wspace=0.3)
# Select a representative experiment (middle one)
rep_idx = len(experiment_results) // 2
rep_result = experiment_results[rep_idx]
steps = np.arange(len(rep_result["stress_history"]))
# ========== SOC PROPERTY 1: SLOW DRIVE ==========
ax1 = fig.add_subplot(gs[0, 0])
ax1.plot(steps, rep_result["param_history"], 'purple', linewidth=2.5)
ax1.axvline(rep_result["n_equilibration"], color='red', linestyle='--',
linewidth=2, alpha=0.7, label='Perturbation start')
ax1.fill_between(steps[:rep_result["n_equilibration"]],
0, 0.3, alpha=0.1, color='blue')
ax1.fill_between(steps[rep_result["n_equilibration"]:],
0, 0.3, alpha=0.15, color='red')
ax1.set_xlabel('Time Step', fontsize=11, fontweight='bold')
ax1.set_ylabel('Predator Death Rate', fontsize=11, fontweight='bold')
ax1.set_title('1) SLOW DRIVE\nGradual Parameter Change',
fontsize=12, fontweight='bold', color='darkblue')
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)
# ========== SOC PROPERTY 2: BUILD-UP OF STRESS ==========
ax2 = fig.add_subplot(gs[0, 1])
ax2.plot(steps, rep_result["stress_history"], 'b-', linewidth=2.5, label='Stress Level')
ax2.axvline(rep_result["n_equilibration"], color='red', linestyle='--',
linewidth=2, alpha=0.7, label='Perturbation start')
# Mark avalanche events with stars
for event_t, event_mag in rep_result["avalanches"]:
ax2.scatter(event_t, rep_result["stress_history"][event_t],
color='orange', s=150, marker='*', zorder=5, edgecolors='black', linewidth=1.5)
ax2.set_xlabel('Time Step', fontsize=11, fontweight='bold')
ax2.set_ylabel('Stress (Interface Density)', fontsize=11, fontweight='bold')
ax2.set_title('2) BUILD-UP OF STRESS\nThresholds & Potential Energy',
fontsize=12, fontweight='bold', color='darkblue')
ax2.legend(fontsize=10, loc='upper left')
ax2.grid(True, alpha=0.3)
# ========== SOC PROPERTY 3: INTERMITTENT RELEASE ==========
ax3 = fig.add_subplot(gs[1, 0])
prey = rep_result["prey_populations"]
pred = rep_result["pred_populations"]
ax3_twin = ax3.twinx()
line1 = ax3.plot(steps, prey, 'g-', label='Prey', linewidth=2.5)
line2 = ax3_twin.plot(steps, pred, 'r-', label='Predator', linewidth=2.5)
ax3.axvline(rep_result["n_equilibration"], color='gray', linestyle='--',
alpha=0.6, linewidth=1.5)
ax3.set_xlabel('Time Step', fontsize=11, fontweight='bold')
ax3.set_ylabel('Prey Population', color='g', fontsize=11, fontweight='bold')
ax3_twin.set_ylabel('Predator Population', color='r', fontsize=11, fontweight='bold')
ax3.set_title('3) INTERMITTENT RELEASE\nAvalanche Cascades',
fontsize=12, fontweight='bold', color='darkblue')
ax3.tick_params(axis='y', labelcolor='g')
ax3_twin.tick_params(axis='y', labelcolor='r')
ax3.grid(True, alpha=0.3)
# Combine legends
lines = line1 + line2
labels = [l.get_label() for l in lines]
ax3.legend(lines, labels, fontsize=10, loc='upper left')
# ========== SOC PROPERTY 4: SELF-ORGANIZATION ==========
ax4 = fig.add_subplot(gs[1, 1])
# Stress-density relation: shows universal behavior across configurations
densities_list = []
stresses_list = []
avalanche_counts_list = []
for result in experiment_results:
# Calculate mean population density during observation phase
prey_pop = result["prey_populations"][result["n_equilibration"]:]
pred_pop = result["pred_populations"][result["n_equilibration"]:]
total_pop = (prey_pop + pred_pop).mean()
grid_size = result["config"]["rows"] * result["config"]["cols"]
density = total_pop / grid_size
# Mean stress during observation phase
mean_stress = result["stress_history"][result["n_equilibration"]:].mean()
avalanche_count = len(result["avalanches"])
densities_list.append(density)
stresses_list.append(mean_stress)
avalanche_counts_list.append(avalanche_count)
# Scatter plot: stress vs density, colored by avalanche activity
scatter = ax4.scatter(densities_list, stresses_list, c=avalanche_counts_list,
cmap='plasma', s=300, alpha=0.8, edgecolors='none')
ax4.set_xlabel('Population Density', fontsize=11, fontweight='bold')
ax4.set_ylabel('Mean Stress Level', fontsize=11, fontweight='bold')
ax4.set_title('4) SELF-ORGANIZATION\nStress-Density Relation',
fontsize=12, fontweight='bold', color='darkblue')
cbar = plt.colorbar(scatter, ax=ax4)
cbar.set_label('Avalanche Count', fontsize=10, fontweight='bold')
ax4.grid(True, alpha=0.3)
plt.suptitle('Prey-Predator Cellular Automaton: Four SOC Properties',
fontsize=14, fontweight='bold', y=0.98)
if output_file:
plt.savefig(output_file, dpi=150, bbox_inches='tight')
print(f"Visualization saved to {output_file}")
return fig
# ============================================================================
# 6. MAIN EXPERIMENT
# ============================================================================
def main():
"""Run complete SOC analysis."""
print("=" * 80)
print("SELF-ORGANIZED CRITICALITY ANALYSIS: Prey-Predator Cellular Automaton")
print("=" * 80)
print()
# Generate diverse parameter configurations
print("[1/4] Generating parameter configurations...")
n_configs = 8 # Small sample for demonstration (not full analysis)
configs = sample_parameter_configurations(n_samples=n_configs, base_seed=42)
print(f" Generated {n_configs} configurations with varied:")
print(" - Grid sizes (16x16 to 64x64)")
print(" - Initial densities (prey: 0.1-0.4, pred: 0.02-0.15)")
print(" - Neighborhoods (Neumann/Moore)")
print(" - Synchronicity (sync/async)")
print(" - Rate parameters (beyond just death/birth)")
print()
# Run perturbation experiments
print("[2/4] Running perturbation experiments...")
experiment_results = []
for i, config in enumerate(configs):
print(f" Config {i+1}/{n_configs}: "
f"grid={config['rows']}x{config['cols']}, "
f"densities=({config['densities'][0]:.2f},{config['densities'][1]:.2f}), "
f"sync={config['synchronous']}")
result = run_soc_perturbation_experiment(
config,
n_equilibration=80, # build stress without perturbation
n_observation=150, # observe cascades during/after perturbation
perturbation_step=80
)
experiment_results.append(result)
print(f" Completed {len(experiment_results)} experiments")
print()
# Analyze robustness
print("[3/4] Analyzing SOC robustness across configurations...")
robustness_metrics = analyze_soc_robustness(experiment_results)
print(f" Avalanche count (avg): {robustness_metrics['avg_avalanche_count']:.2f} "
f"(std: {robustness_metrics['std_avalanche_count']:.2f})")
print(f" Avalanche magnitude (avg): {robustness_metrics['avalanche_magnitude_mean']:.4f}")
print(f" Stress level (avg): {robustness_metrics['avg_stress']:.4f}")
print(f" Coefficient of Variation (avalanche count): {robustness_metrics['coefficient_of_variation_avalanche']:.3f}")
if robustness_metrics['coefficient_of_variation_avalanche'] < 1.0:
print(" → LOW variation indicates ROBUST criticality across diverse parameters ✓")
else:
print(" → HIGH variation indicates parameter-dependent behavior")
print()
# Create visualization
print("[4/4] Creating comprehensive visualization...")
output_path = Path(__file__).parent.parent / "soc_analysis_results.png"
visualize_soc_properties(experiment_results, robustness_metrics, str(output_path))
print(f" Saved to: {output_path}")
if __name__ == "__main__":
main()