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

// 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

{
  "type": "ObDecl",
  "name": "MarketShare",
  "theory_type": "Variable",
  "description": "company's percentage of total market"
}

Positive Link

{
  "type": "MorDecl",
  "name": "growth_effect",
  "dom": "MarketShare",
  "cod": "Revenue",
  "theory_type": "Positive",
  "description": "higher market share increases revenue"
}

Negative Link

{
  "type": "MorDecl",
  "name": "saturation_effect",
  "dom": "MarketShare",
  "cod": "GrowthRate",
  "theory_type": "Negative",
  "description": "higher share reduces growth potential"
}

Delay (CatColab 0.2)

{
  "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

# 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

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Skill Name: catcolab-causal-loop Type: Systems Dynamics / Feedback Analysis Trit: 0 (ERGODIC) GF(3): Conserved via triadic composition