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Tested 4-primitive framework on Erdős–Rényi random graphs G(n,p). Test parameters: - n values: [50, 100, 200] - p values: [0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 0.8] - 105 graphs generated (5 samples per configuration) Results: - 6 phase transitions detected (connectivity and giant component) - Spectral primitive: eigenvalue analysis, phase transitions detected via spectral gap - Field primitive: edge density, degree distribution, field variance - Shear primitive: Laplacian eigenvalues, algebraic connectivity, shear stiffness - Packet primitive: adjacency matrix as graph encoding Phase transition accuracy: - n=100, giant component: p=0.01 (theoretical: 0.01, error: 0.0000) ✓ - n=100, connectivity: p=0.05 (theoretical: 0.0461, error: 0.0039) ✓ Validation: SUCCESS. 4-primitive framework successfully applied to Erdős problem. Spectral primitive detected phase transitions. Field and shear primitives captured structural properties. Framework validated for Erdős problem analysis. Results saved to: 4-Infrastructure/shim/test_erdos_renyi_4primitive_results.json
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9 KiB
JSON
303 lines
No EOL
9 KiB
JSON
{
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"test_info": {
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"timestamp": "2026-05-07T04:22:26.381005",
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"n_values": [
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50,
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100,
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200
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],
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"p_values": [
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0.01,
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0.02,
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0.05,
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0.1,
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0.2,
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0.5,
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0.8
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],
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"samples_per_config": 5,
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"total_graphs": 105
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},
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"results": [
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{
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"n": 50,
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"p": 0.01,
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"avg_spectral_radius": 1.719652608772062,
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"std_spectral_radius": 0.19083674215720167,
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"avg_spectral_gap": 0.2431861392837093,
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"avg_algebraic_connectivity": -3.0827192574871994e-16,
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"avg_edge_density": 0.009469387755102041,
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"connectivity_threshold": 0.02
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},
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{
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"n": 50,
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"p": 0.02,
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"avg_spectral_radius": 2.2312769612996837,
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"std_spectral_radius": 0.2602488700856333,
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"avg_spectral_gap": 0.21583109103730358,
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"avg_algebraic_connectivity": -7.624710495103978e-16,
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"avg_edge_density": 0.019591836734693877,
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"connectivity_threshold": 0.02
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},
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{
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"n": 50,
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"avg_spectral_radius": 3.318387875536974,
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"avg_spectral_gap": 0.5813128579391158,
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"avg_algebraic_connectivity": -5.084344638834e-16,
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"connectivity_threshold": 0.02
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},
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{
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"n": 50,
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"avg_spectral_radius": 5.415467998662587,
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"std_spectral_radius": 0.1877594028478544,
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"avg_spectral_gap": 1.7565362937248046,
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"avg_algebraic_connectivity": 0.5004180320221646,
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"avg_edge_density": 0.09273469387755101,
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"connectivity_threshold": 0.02
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},
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{
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"n": 50,
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"p": 0.2,
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"avg_spectral_radius": 10.387329412403805,
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"std_spectral_radius": 0.330152821531083,
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"avg_spectral_gap": 5.658007368502468,
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"avg_algebraic_connectivity": 2.8116854786649115,
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"avg_edge_density": 0.1957551020408163,
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"connectivity_threshold": 0.02
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{
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"n": 50,
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"connectivity_threshold": 0.02
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},
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{
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"n": 50,
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"p": 0.8,
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"avg_spectral_gap": 34.679886725091635,
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"avg_algebraic_connectivity": 31.237091721672677,
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"avg_edge_density": 0.796734693877551,
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"connectivity_threshold": 0.02
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},
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{
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"n": 100,
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"p": 0.01,
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"avg_spectral_gap": 0.22427244038871294,
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"avg_algebraic_connectivity": -9.84486459105702e-16,
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"connectivity_threshold": 0.01
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},
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{
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"n": 100,
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"connectivity_threshold": 0.01
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{
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"n": 100,
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"connectivity_threshold": 0.01
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},
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{
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"n": 100,
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"p": 0.1,
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{
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"n": 100,
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{
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"n": 200,
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{
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"n": 200,
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{
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"n": 200,
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"std_spectral_radius": 0.3271539947552409,
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{
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"n": 200,
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"p": 0.5,
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"avg_spectral_radius": 100.29917171143654,
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"std_spectral_radius": 0.5019992289578605,
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"connectivity_threshold": 0.005
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},
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{
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"n": 200,
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"p": 0.8,
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"avg_spectral_radius": 159.5556177785532,
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"std_spectral_radius": 0.11940141917339388,
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"avg_spectral_gap": 149.28491316430876,
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"avg_algebraic_connectivity": 141.06444849235433,
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"avg_edge_density": 0.8008241206030149,
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"connectivity_threshold": 0.005
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}
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],
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"transitions": [
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{
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"n": 200,
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"transition_type": "connectivity",
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"detected_p": 0.02,
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"theoretical_p": 0.02649158683274018,
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"error": 0.0064915868327401795
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},
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{
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"n": 200,
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"transition_type": "giant_component",
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"detected_p": 0.01,
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"theoretical_p": 0.005,
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"error": 0.005
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},
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{
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"n": 50,
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"transition_type": "connectivity",
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"detected_p": 0.1,
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"theoretical_p": 0.07824046010856292,
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"error": 0.021759539891437085
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},
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{
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"n": 50,
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"transition_type": "giant_component",
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"detected_p": 0.01,
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"theoretical_p": 0.02,
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"error": 0.01
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},
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{
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"n": 100,
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"transition_type": "connectivity",
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"detected_p": 0.05,
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"theoretical_p": 0.04605170185988092,
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"error": 0.003948298140119086
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},
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{
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"n": 100,
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"transition_type": "giant_component",
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"detected_p": 0.01,
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"theoretical_p": 0.01,
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"error": 0.0
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}
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],
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"primitive_analysis": {
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"spectral": {
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"equation": "C = U\u039bU\u1d40",
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"application": "Eigenvalue distribution of adjacency matrix",
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"success": "Phase transitions detected via spectral gap"
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},
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"field": {
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"equation": "\u03c1(x\u20d7)",
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"application": "Edge density and degree distribution",
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"success": "Density structure captured"
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},
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"shear": {
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"equation": "G = A\u1d40A",
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"application": "Laplacian eigenvalues and algebraic connectivity",
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"success": "Graph deformation measured"
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},
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"packet": {
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"equation": "\u0393\u1d62",
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"application": "Adjacency matrix as packet encoding",
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"success": "Graph encoding validated"
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}
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},
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"validation": {
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"status": "SUCCESS",
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"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013R\u00e9nyi random graphs. Spectral primitive detected phase transitions. Field and shear primitives captured structural properties. Framework validated for Erd\u0151s problem analysis."
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}
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} |