catcolab-causal-loop

# CatColab Causal Loop Diagrams: Systems Dynamics **Trit**: 0 (ERGODIC - coordinator/mediator) **Color**: Yellow (#FFD700) ## Overview Causal Loop Diagrams (CLDs) in CatColab model feedback systems: - **Variables**: System quantities that change over time - **Positive links (+)**: Same-direction influence (increase→increase) - **Negative links (-)**: Opposite-direction influence (increase→decrease) - **Loops**: Reinforcing (R) or Balancing (B) feedback CLDs are essential for understanding system behavior, policy analysis, and strategic planning. ## Mathematical Foundation A causal loop diagram is a **signed directed graph** with loop classification: ``` ┌─────────────────────────────────────────────────────┐ │ CAUSAL LOOP DIAGRAM │ ├─────────────────────────────────────────────────────┤ │ Variables: │ │ Population, Resources, Pollution, Quality │ │ │ │ Positive Links (+): │ │ Population ──(+)──► Pollution │ │ Resources ──(+)──► Quality │ │ │ │ Negative Links (-): │ │ Pollution ──(-)──► Quality │ │ Quality ──(-)──► Population (emigration) │ │ │ │ Loops: │ │ R1: Population→Births→Population (reinforcing) │ │ B1: Population→Resources→Quality→Pop (balancing) │ └─────────────────────────────────────────────────────┘ ``` ## Loop Classification **Reinforcing Loop (R)**: Even number of negative links - Exponential growth or collapse - "Snowball effect" or "vicious/virtuous cycle" **Balancing Loop (B)**: Odd number of negative links - Goal-seeking behavior - Homeostasis, equilibrium ``` REINFORCING (R): BALANCING (B): A ──(+)──► B A ──(+)──► B ▲ │ ▲ │ │ │ │ │ └──(+)─────┘ └──(-)─────┘ (exponential) (equilibrium) ``` ## Double Theory ```rust // Causal loop double theory with decorated edges pub fn th_causal_loop() -> DiscreteDblTheory { let mut cat = FpCategory::new(); // Object type cat.add_ob_generator(name("Variable")); // Morphism types (polarized links) cat.add_mor_generator(name("Positive"), name("Variable"), name("Variable")); cat.add_mor_generator(name("Negative"), name("Variable"), name("Variable")); // Decorations (CatColab 0.2) cat.add_mor_generator(name("Delay"), name("Variable"), name("Variable")); cat.add_mor_generator(name("Indeterminate"), name("Variable"), name("Variable")); cat.into() } ``` ## CatColab Implementation ### Variable Declaration ```typescript { "type": "ObDecl", "name": "MarketShare", "theory_type": "Variable", "description": "company's percentage of total market" } ``` ### Positive Link ```typescript { "type": "MorDecl", "name": "growth_effect", "dom": "MarketShare", "cod": "Revenue", "theory_type": "Positive", "description": "higher market share increases revenue" } ``` ### Negative Link ```typescript { "type": "MorDecl", "name": "saturation_effect", "dom": "MarketShare", "cod": "GrowthRate", "theory_type": "Negative", "description": "higher share reduces growth potential" } ``` ### Delay (CatColab 0.2) ```typescript { "type": "MorDecl", "name": "investment_lag", "dom": "RnD_Spending", "cod": "ProductQuality", "theory_type": "Delay", "delay_time": 12, // months "description": "R&D takes time to improve products" } ``` ## Lotka-Volterra Semantics CatColab generates **Lotka-Volterra ODEs** from causal loops: ``` For variables X, Y with positive link X→Y: dY/dt = α·X·Y For negative link X→Y: dY/dt = -β·X·Y General form: dXᵢ/dt = Xᵢ · Σⱼ aᵢⱼ·Xⱼ ``` ## Practical Examples ### Example 1: Adoption Dynamics ``` Word of Mouth ↗ (+) Users ────────► Adoption Rate ▲ │ │ │ └────(+)──────────┘ R1: Viral Growth Adoption Rate ──(+)──► Users │ └──(-)──► Potential Users │ B1: Market Saturation ``` ### Example 2: Thermostat (Balancing) ``` Desired Temp ──(+)──► Gap ▲ │ │ │ │ (+) │ ▼ Actual Temp ◄──(+)── Heating │ └──(-)──► Gap B1: Temperature Control ``` ### Example 3: Arms Race (Reinforcing) ``` Country A Arms ──(+)──► Country A Threat Perception ▲ │ │ (+) │ ▼ Country B Arms ◄──(+)── Country B Arms Spending │ └──(+)──► Country A Threat Perception R1: Escalation Spiral ``` ## Analysis Capabilities - **Loop identification**: Automatic detection of R and B loops - **Dominant loop analysis**: Which loops drive behavior - **Policy leverage points**: Where interventions are most effective - **Scenario simulation**: Lotka-Volterra dynamics ## GF(3) Triads ``` catcolab-regulatory-networks (-1) ⊗ catcolab-causal-loop (0) ⊗ catcolab-stock-flow (+1) = 0 ✓ open-games (-1) ⊗ catcolab-causal-loop (0) ⊗ dynamical-system-functor (+1) = 0 ✓ ``` ## Commands ```bash # Create causal loop diagram just catcolab-new causal-loop "market-dynamics" # Identify all loops just catcolab-analyze market-dynamics --loops # Simulate Lotka-Volterra just catcolab-simulate market-dynamics --lotka-volterra # Export to Vensim format just catcolab-export market-dynamics --format=mdl ``` ## References - Sterman (2000) "Business Dynamics: Systems Thinking and Modeling" - Meadows (2008) "Thinking in Systems" - [CatColab Causal Loop Help](https://catcolab.org/help/logics/causal-loop) --- **Skill Name**: catcolab-causal-loop **Type**: Systems Dynamics / Feedback Analysis **Trit**: 0 (ERGODIC) **GF(3)**: Conserved via triadic composition