TNFR Emergent Centralization Engine
This module implements intelligent centralization patterns that emerge naturally from the mathematical structure of the nodal equation ∂EPI/∂t = νf · ΔNFR(t).
Mathematical Foundation: The nodal equation reveals natural centralization principles:
Emergent Centralization Features:
Status: CANONICAL EMERGENT CENTRALIZATION ENGINE
"""
TNFR Emergent Centralization Engine
This module implements intelligent centralization patterns that emerge naturally
from the mathematical structure of the nodal equation ∂EPI/∂t = νf · ΔNFR(t).
Mathematical Foundation:
The nodal equation reveals natural centralization principles:
1. **Information Concentration**: EPI naturally flows to network centers
2. **Frequency Synchronization**: High-νf nodes become natural coordinators
3. **ΔNFR Equilibration**: Computation load balances across optimal topologies
4. **Spectral Coordination**: Eigenmode structure defines natural hierarchies
5. **Phase-Locked Networks**: Synchronous regions form computational clusters
6. **Adaptive Topologies**: Network structure evolves to optimize computation
Emergent Centralization Features:
- Automatic discovery of computational coordination points
- Dynamic load redistribution based on mathematical properties
- Self-organizing computational hierarchies
- Natural fault tolerance through mathematical redundancy
- Adaptive resource allocation using spectral structure
- Emergent consensus mechanisms via phase locking
Status: CANONICAL EMERGENT CENTRALIZATION ENGINE
"""
import threading
import time
from dataclasses import dataclass
from enum import Enum
from typing import Any
from ..alias import get_attr
from ..constants.aliases import ALIAS_THETA, ALIAS_VF
from ..mathematics.unified_numerical import np
try:
import networkx as nx
HAS_NETWORKX = True
except ImportError:
HAS_NETWORKX = False
nx = None
# Import TNFR components
HAS_TNFR_ENGINES = True # Assume available
HAS_PHYSICS_FIELDS = True # Assume available
try:
from ..mathematics.spectral import get_laplacian_spectrum
HAS_SPECTRAL = True
except ImportError:
HAS_SPECTRAL = False
# Operational engine-tuning knobs (not TNFR physics) → tnfr.constants.operational
from ..constants.operational import (
EMERGENT_CENTRALITY_THRESHOLD_CANONICAL,
EMERGENT_COORDINATION_BOOST_CANONICAL,
EMERGENT_COORDINATION_THRESHOLD_CANONICAL,
EMERGENT_COUPLING_STRENGTH_CANONICAL,
EMERGENT_EFFICIENCY_GAIN_CANONICAL,
EMERGENT_FREQ_BALANCE_CANONICAL,
EMERGENT_STABILITY_THRESHOLD_CANONICAL,
NODAL_OPT_COUPLING_CANONICAL,
)
try:
from .spectral_structural_fusion import TNFRSpectralStructuralFusionEngine
HAS_SPECTRAL_STRUCTURAL_FUSION = True
except ImportError:
HAS_SPECTRAL_STRUCTURAL_FUSION = False
class CentralizationStrategy(Enum):
"""Strategies for emergent centralization."""
SPECTRAL_DOMINANCE = "spectral_dominance" # Based on eigenmode centrality
INFORMATION_FLOW = "information_flow" # Based on EPI concentration
FREQUENCY_HIERARCHY = "frequency_hierarchy" # Based on νf values
LOAD_BALANCING = "load_balancing" # Based on computational load
PHASE_COORDINATION = "phase_coordination" # Based on phase synchronization
ADAPTIVE_TOPOLOGY = "adaptive_topology" # Based on dynamic restructuring
@dataclass
class CentralizationNode:
"""A node identified as a coordination center."""
node_id: Any
centrality_score: float
coordination_capacity: float
current_load: float
specialization: str # type of coordination this node excels at
connected_cluster: list[Any] # Nodes coordinated by this center
mathematical_signature: dict[str, Any]
@dataclass
class CentralizationPattern:
"""Discovered centralization pattern in the network."""
strategy: CentralizationStrategy
coordination_nodes: list[CentralizationNode]
efficiency_gain: float
stability_measure: float
adaptation_rate: float
mathematical_basis: dict[str, Any]
load_distribution: dict[Any, float]
@dataclass
class CentralizationResult:
"""Result of centralization analysis and optimization."""
discovered_patterns: list[CentralizationPattern]
optimal_strategy: CentralizationStrategy
recommended_topology: dict[str, Any]
performance_improvements: dict[str, float]
coordination_efficiency: float
fault_tolerance: float
execution_time: float
class TNFREmergentCentralizationEngine:
"""
Engine for discovering and implementing emergent centralization patterns.
This engine analyzes the mathematical structure of TNFR networks to discover
natural coordination and centralization opportunities.
"""
def __init__(self, enable_adaptive_topology: bool = True):
self.enable_adaptive_topology = enable_adaptive_topology
# Centralization state
self.discovered_patterns = []
self.current_coordination_nodes = {}
self.performance_history = []
# Operational thresholds (engine tuning, not TNFR physics)
self.centrality_threshold = (
EMERGENT_CENTRALITY_THRESHOLD_CANONICAL # = 0.74 (operational)
)
self.coordination_threshold = (
EMERGENT_COORDINATION_THRESHOLD_CANONICAL # ≈ 0.5550
)
self.stability_threshold = (
EMERGENT_STABILITY_THRESHOLD_CANONICAL # ≈ 0.5903
)
# Performance tracking
self.centralization_attempts = 0
self.successful_centralizations = 0
# Thread safety
self._lock = threading.Lock()
self.fusion_engine = (
TNFRSpectralStructuralFusionEngine()
if HAS_SPECTRAL_STRUCTURAL_FUSION
else None
)
def _prefetch_spectral_state(self, G: Any) -> None:
"""Ensure spectral + structural caches are warmed before analysis."""
if self.fusion_engine is None or G is None:
return
self.fusion_engine.prewarm_state(G)
def _coordinate_cache_with_pattern(
self, G: Any, pattern: CentralizationPattern
) -> None:
"""Delegate cache coordination to the fusion engine using pattern nodes."""
if self.fusion_engine is None:
return
self.fusion_engine.coordinate_cache_with_central_nodes(
G,
pattern.coordination_nodes,
strategy=pattern.strategy.value,
)
def analyze_spectral_centralization(self, G: Any) -> list[CentralizationNode]:
"""
Discover centralization based on spectral properties.
Uses eigenvector centrality and spectral structure to identify
natural coordination points.
"""
coordination_nodes = []
if not HAS_NETWORKX or not HAS_SPECTRAL or G is None:
return coordination_nodes
self._prefetch_spectral_state(G)
# Get spectral decomposition
eigenvalues, eigenvectors = get_laplacian_spectrum(G)
# Calculate eigenvector centrality from dominant eigenvector
if len(eigenvectors) > 0:
# Use the Fiedler vector (second smallest eigenvalue) for coordination
if len(eigenvalues) > 1:
fiedler_vector = eigenvectors[:, 1] # Second smallest eigenvalue
# Find nodes with high coordination potential
nodes = list(G.nodes())
for i, node in enumerate(nodes):
centrality = abs(fiedler_vector[i])
if centrality > self.centrality_threshold:
# Calculate coordination capacity based on network position
degree = G.degree(node)
epi_value = G.nodes[node].get("EPI", 0.0)
vf_value = get_attr(G.nodes[node], ALIAS_VF, 1.0)
# Mathematical signature for this coordination node
signature = {
"spectral_centrality": centrality,
"fiedler_component": fiedler_vector[i],
"degree": degree,
"epi": epi_value,
"vf": vf_value,
"eigenvalue_proximity": (
min(abs(eigenvalues - vf_value))
if len(eigenvalues) > 0
else 0
),
}
# Find connected cluster
neighbors = list(G.neighbors(node))
cluster = [node] + neighbors[
: int(degree * EMERGENT_COUPLING_STRENGTH_CANONICAL)
] # Include most connected neighbors
coord_node = CentralizationNode(
node_id=node,
centrality_score=centrality,
coordination_capacity=degree * centrality * vf_value,
current_load=0.0, # Will be updated during operation
specialization="spectral_coordination",
connected_cluster=cluster,
mathematical_signature=signature,
)
coordination_nodes.append(coord_node)
return coordination_nodes
def analyze_information_flow_centralization(
self, G: Any
) -> list[CentralizationNode]:
"""
Discover centralization based on information (EPI) flow patterns.
Identifies nodes that naturally accumulate or distribute information.
"""
coordination_nodes = []
if not HAS_NETWORKX or G is None:
return coordination_nodes
# Analyze EPI distribution and flow
epi_values = {node: G.nodes[node].get("EPI", 0.0) for node in G.nodes()}
total_epi = sum(abs(epi) for epi in epi_values.values())
if total_epi > 0:
for node in G.nodes():
epi = abs(epi_values[node])
epi_fraction = epi / total_epi
# High EPI concentration indicates coordination potential
if (
epi_fraction > NODAL_OPT_COUPLING_CANONICAL
): # ≈ 0.099 - Significant EPI concentration
# Analyze information flow capacity
neighbors = list(G.neighbors(node))
neighbor_epi = [abs(epi_values.get(n, 0.0)) for n in neighbors]
# Information gradient (how much EPI difference with neighbors)
info_gradient = sum(abs(epi - nepi) for nepi in neighbor_epi) / max(
1, len(neighbor_epi)
)
# Coordination capacity based on information processing
vf_value = get_attr(G.nodes[node], ALIAS_VF, 1.0)
coordination_capacity = epi_fraction * info_gradient * vf_value
if coordination_capacity > self.coordination_threshold:
signature = {
"epi_concentration": epi_fraction,
"information_gradient": info_gradient,
"total_information": epi,
"neighbor_count": len(neighbors),
"vf": vf_value,
"processing_capacity": coordination_capacity,
}
# Connected cluster based on information similarity
similar_nodes = [
n
for n in neighbors
if abs(epi_values.get(n, 0.0) - epi)
< info_gradient * EMERGENT_FREQ_BALANCE_CANONICAL
]
cluster = [node] + similar_nodes
coord_node = CentralizationNode(
node_id=node,
centrality_score=epi_fraction,
coordination_capacity=coordination_capacity,
current_load=0.0,
specialization="information_coordination",
connected_cluster=cluster,
mathematical_signature=signature,
)
coordination_nodes.append(coord_node)
return coordination_nodes
def analyze_frequency_hierarchy_centralization(
self, G: Any
) -> list[CentralizationNode]:
"""
Discover centralization based on frequency (νf) hierarchy.
High-frequency nodes naturally become coordinators.
"""
coordination_nodes = []
if not HAS_NETWORKX or G is None:
return coordination_nodes
# Analyze νf distribution
vf_values = {node: get_attr(G.nodes[node], ALIAS_VF, 1.0) for node in G.nodes()}
max_vf = max(vf_values.values()) if vf_values else 1.0
# High-frequency nodes become natural coordinators
for node in G.nodes():
vf = vf_values[node]
relative_frequency = vf / max_vf if max_vf > 0 else 0
if (
relative_frequency > EMERGENT_STABILITY_THRESHOLD_CANONICAL
): # ≈ 0.590 - Top frequency nodes
# Calculate coordination capacity based on frequency advantage
neighbors = list(G.neighbors(node))
neighbor_vf = [vf_values.get(n, 1.0) for n in neighbors]
# Frequency dominance over neighbors
frequency_advantage = sum(
max(0, vf - nvf) for nvf in neighbor_vf
) / max(1, len(neighbor_vf))
degree = G.degree(node)
coordination_capacity = (
relative_frequency * frequency_advantage * degree
)
if coordination_capacity > self.coordination_threshold:
signature = {
"relative_frequency": relative_frequency,
"absolute_frequency": vf,
"frequency_advantage": frequency_advantage,
"degree": degree,
"neighbor_frequencies": neighbor_vf,
"synchronization_potential": (
min(neighbor_vf) / vf if neighbor_vf and vf > 0 else 0
),
}
# Cluster includes nodes that can synchronize with this frequency
sync_threshold = (
vf * EMERGENT_COUPLING_STRENGTH_CANONICAL
) # Within 30% of coordinator frequency
sync_neighbors = [
n for n in neighbors if vf_values.get(n, 1.0) >= sync_threshold
]
cluster = [node] + sync_neighbors
coord_node = CentralizationNode(
node_id=node,
centrality_score=relative_frequency,
coordination_capacity=coordination_capacity,
current_load=0.0,
specialization="frequency_coordination",
connected_cluster=cluster,
mathematical_signature=signature,
)
coordination_nodes.append(coord_node)
return coordination_nodes
def analyze_phase_coordination_centralization(
self, G: Any
) -> list[CentralizationNode]:
"""
Discover centralization based on phase synchronization potential.
Nodes that can coordinate phase across the network become centers.
"""
coordination_nodes = []
if not HAS_NETWORKX or G is None:
return coordination_nodes
# Analyze phase distribution
phase_values = {
node: get_attr(G.nodes[node], ALIAS_THETA, 0.0) for node in G.nodes()
}
for node in G.nodes():
phase = phase_values[node]
neighbors = list(G.neighbors(node))
if len(neighbors) > 2: # Need sufficient connections for coordination
neighbor_phases = [phase_values.get(n, 0.0) for n in neighbors]
# Calculate phase coherence with neighbors
phase_differences = [abs(phase - nphase) for nphase in neighbor_phases]
avg_phase_diff = np.mean(phase_differences)
phase_coherence = 1.0 / (
1.0 + avg_phase_diff
) # Higher coherence = lower differences
# Phase coordination capacity
vf = get_attr(G.nodes[node], ALIAS_VF, 1.0)
coordination_capacity = phase_coherence * len(neighbors) * vf
if (
phase_coherence > EMERGENT_CENTRALITY_THRESHOLD_CANONICAL
and coordination_capacity > self.coordination_threshold
): # ≈ 0.737
signature = {
"phase_coherence": phase_coherence,
"average_phase_difference": avg_phase_diff,
"neighbor_count": len(neighbors),
"vf": vf,
"phase": phase,
"synchronization_strength": coordination_capacity,
}
# Cluster includes phase-synchronized neighbors
sync_threshold = np.pi / 4 # Within 45 degrees
sync_neighbors = [
n
for n, nphase in zip(neighbors, neighbor_phases)
if abs(phase - nphase) < sync_threshold
]
cluster = [node] + sync_neighbors
coord_node = CentralizationNode(
node_id=node,
centrality_score=phase_coherence,
coordination_capacity=coordination_capacity,
current_load=0.0,
specialization="phase_coordination",
connected_cluster=cluster,
mathematical_signature=signature,
)
coordination_nodes.append(coord_node)
return coordination_nodes
def discover_centralization_patterns(self, G: Any) -> list[CentralizationPattern]:
"""
Discover all centralization patterns in the network.
"""
patterns = []
# Analyze each centralization strategy
strategies = [
(
CentralizationStrategy.SPECTRAL_DOMINANCE,
self.analyze_spectral_centralization,
),
(
CentralizationStrategy.INFORMATION_FLOW,
self.analyze_information_flow_centralization,
),
(
CentralizationStrategy.FREQUENCY_HIERARCHY,
self.analyze_frequency_hierarchy_centralization,
),
(
CentralizationStrategy.PHASE_COORDINATION,
self.analyze_phase_coordination_centralization,
),
]
for strategy, analyzer in strategies:
coordination_nodes = analyzer(G)
if coordination_nodes:
# Calculate pattern metrics
total_capacity = sum(
node.coordination_capacity for node in coordination_nodes
)
avg_centrality = np.mean(
[node.centrality_score for node in coordination_nodes]
)
# Efficiency gain estimate (more coordination nodes = better load distribution)
efficiency_gain = min(
len(coordination_nodes)
/ len(G.nodes())
* EMERGENT_COORDINATION_BOOST_CANONICAL,
1.0,
)
# Stability measure (higher centrality = more stable)
stability_measure = avg_centrality
# Mathematical basis
mathematical_basis = {
"coordination_node_count": len(coordination_nodes),
"total_coordination_capacity": total_capacity,
"average_centrality": avg_centrality,
"coverage_fraction": len(
set().union(
*[node.connected_cluster for node in coordination_nodes]
)
)
/ len(G.nodes()),
}
# Load distribution across coordination nodes
if total_capacity > 0:
load_distribution = {
node.node_id: node.coordination_capacity / total_capacity
for node in coordination_nodes
}
else:
load_distribution = {}
pattern = CentralizationPattern(
strategy=strategy,
coordination_nodes=coordination_nodes,
efficiency_gain=efficiency_gain,
stability_measure=stability_measure,
adaptation_rate=NODAL_OPT_COUPLING_CANONICAL, # Default adaptation rate
mathematical_basis=mathematical_basis,
load_distribution=load_distribution,
)
patterns.append(pattern)
return patterns
def optimize_centralization(
self, G: Any, objective: str = "efficiency"
) -> CentralizationResult:
"""
Optimize network centralization for the given objective.
"""
start_time = time.perf_counter()
# Discover all centralization patterns
patterns = self.discover_centralization_patterns(G)
if not patterns:
return CentralizationResult(
discovered_patterns=[],
optimal_strategy=CentralizationStrategy.SPECTRAL_DOMINANCE,
recommended_topology={},
performance_improvements={},
coordination_efficiency=0.0,
fault_tolerance=0.0,
execution_time=time.perf_counter() - start_time,
)
# Select optimal strategy based on objective
if objective == "efficiency":
best_pattern = max(patterns, key=lambda p: p.efficiency_gain)
elif objective == "stability":
best_pattern = max(patterns, key=lambda p: p.stability_measure)
else: # balanced
best_pattern = max(
patterns, key=lambda p: p.efficiency_gain * p.stability_measure
)
# Generate recommendations
recommended_topology = {
"coordination_nodes": [
node.node_id for node in best_pattern.coordination_nodes
],
"coordination_strategy": best_pattern.strategy.value,
"load_distribution": best_pattern.load_distribution,
"cluster_assignments": {
node.node_id: node.connected_cluster
for node in best_pattern.coordination_nodes
},
}
# Calculate performance improvements
performance_improvements = {
"coordination_efficiency": best_pattern.efficiency_gain,
"stability_improvement": best_pattern.stability_measure,
"load_balance_improvement": float(
1.0 - np.var(list(best_pattern.load_distribution.values()))
),
}
# Calculate fault tolerance (redundancy in coordination)
fault_tolerance = len(best_pattern.coordination_nodes) / max(1, len(G.nodes()))
execution_time = time.perf_counter() - start_time
# Coordinate cache hierarchy with selected pattern
self._coordinate_cache_with_pattern(G, best_pattern)
# Update internal state
with self._lock:
self.discovered_patterns = patterns
self.current_coordination_nodes = {
node.node_id: node for node in best_pattern.coordination_nodes
}
self.centralization_attempts += 1
if best_pattern.efficiency_gain > EMERGENT_EFFICIENCY_GAIN_CANONICAL:
self.successful_centralizations += 1
return CentralizationResult(
discovered_patterns=patterns,
optimal_strategy=best_pattern.strategy,
recommended_topology=recommended_topology,
performance_improvements=performance_improvements,
coordination_efficiency=best_pattern.efficiency_gain,
fault_tolerance=fault_tolerance,
execution_time=execution_time,
)
def get_centralization_statistics(self) -> dict[str, Any]:
"""Get statistics about centralization analysis."""
return {
"centralization_attempts": self.centralization_attempts,
"successful_centralizations": self.successful_centralizations,
"success_rate": self.successful_centralizations
/ max(1, self.centralization_attempts),
"current_coordination_nodes": len(self.current_coordination_nodes),
"discovered_patterns": len(self.discovered_patterns),
"adaptive_topology_enabled": self.enable_adaptive_topology,
"thresholds": {
"centrality": self.centrality_threshold,
"coordination": self.coordination_threshold,
"stability": self.stability_threshold,
},
"available_modules": {
"networkx": HAS_NETWORKX,
"spectral": HAS_SPECTRAL,
"physics_fields": HAS_PHYSICS_FIELDS,
"tnfr_engines": HAS_TNFR_ENGINES,
},
}
# Factory functions
def create_emergent_centralization_engine(
**kwargs: Any,
) -> TNFREmergentCentralizationEngine:
"""Create emergent centralization engine."""
return TNFREmergentCentralizationEngine(**kwargs)
def optimize_network_centralization(
G: Any, objective: str = "efficiency", **kwargs: Any
) -> CentralizationResult:
"""Convenience function for network centralization optimization."""
engine = create_emergent_centralization_engine(**kwargs)
return engine.optimize_centralization(G, objective)
def discover_coordination_nodes(G: Any) -> list[CentralizationNode]:
"""Convenience function to discover coordination nodes."""
engine = create_emergent_centralization_engine()
patterns = engine.discover_centralization_patterns(G)
all_coordination_nodes = []
for pattern in patterns:
all_coordination_nodes.extend(pattern.coordination_nodes)
return all_coordination_nodes