Openai/693b7cce-38f4-800d-92f7-5e56a467bfa7

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User: Below is an academic peer-review & structured research draft responding to your innovative proposal for dynamic NPC per…

Below is an academic peer-review & structured research draft responding to your innovative proposal for dynamic NPC personalities integrating psychological, narrative, and common knowledge frameworks. I will reference real research where available (Pinker’s common knowledge, personality modeling in games, epistemic logic, etc.) to ground the ideas scientifically and conceptually.

---

Dynamic NPC Personalities: A Computational Framework Integrating Narrative, Psychology, and Common Knowledge

1. Introduction

Games increasingly demand NPCs with behaviors and personalities that feel alive and socially grounded. Static scripting and decision trees lack the adaptability and social depth players increasingly expect. This paper outlines an implementable AI architecture where:

NPCs have dynamic personalities that evolve in response to interactions.

Narrative roles (e.g., Hero’s Journey stages, dramatic situations) influence behavior.

Psychological models (Jungian archetypes, OCEAN/MBTI) inform personality vectors.

A common knowledge system ensures shared world understanding that shapes cognition and social coordination.

We propose an architecture combining epistemic logic, psychological representations, and narrative structures into NPC AI that produces emergent, socially plausible behavior.

---

1. Theoretical Foundations

2.1 Narrative and Psychological Models

Hero’s Journey & Dramatic Situations The Hero’s Journey is a widely used narrative schema with identifiable stages that shape character arcs. Dramatic situations (e.g., pursuit, sacrifice) guide conflict and decision contexts.

Jungian Archetypes Archetypes represent prototypical personas (e.g., Hero, Mentor, Shadow). While Jung’s theory is conceptual and not empirically validated, it provides a useful taxonomy for narrative roles. (Psychology consensus: archetypes are metaphors rather than scientifically confirmed constructs.)

Personality Models (MBTI / OCEAN)

MBTI offers categorical personality types widely used in games and design heuristics.

OCEAN/Big Five provides continuous trait measures correlated with real behavior. When integrated into NPC parameters, they allow fine-grained variation in responses (e.g., high Agreeableness → cooperative dialogue).

Empirical work has shown that models such as the Big Five can be used to influence game narrative generation and player experience.

2.2 Common Knowledge in Multi-Agent Systems

Epistemic Logic and Common Knowledge In logic/game theory, common knowledge means a fact is known by all, and each knows that all know it, recursively ad infinitum. This is distinct from mere shared or mutual knowledge where agents know something without knowing that others know it.

Pinker’s Perspective Steven Pinker emphasizes that common knowledge fuels coordination and shared expectations in human social life — enabling coordination, norms, and indirect communication to function.

Why This Matters in NPCs In games with social systems, NPCs need:

A baseline world model (environment, social norms, economic rules).

Recognition that others know what they know to coordinate trust, conflict, cooperation, and deception.

This is what we operationalize as a recursive depth-bounded common knowledge representation.

---

1. Computational Architecture

3.1 Vectorized Personality & Narrative Representations

Each NPC maintains internal vectors for:

Component Representation

Personality OCEAN Continuous Vector Archetype Narrative role vector MBTI Categorical/embedded encoding Emotion Radial affective state (e.g., emotion wheel) Knowledge State Epistemic state with recursion depth

These vectors are used as inputs to decision engines (e.g., neural policies or LLM inference), determining:

Dialogue generation

Social behavior

Narrative progression response

---

3.2 Knowledge State Representation

We define CommonKnowledgeItem as:

{

 "id": "fact_identifier",
 "truth_value": bool,
 "public_visibility": float,
 "public_events": {...},
 "recursion_depth": 3–5

}

Each NPC tracks:

Personal beliefs

Assumed beliefs of others up to a depth

Public vs. private knowledge

This data feeds into epistemic reasoning about what others know and expect.

3.3 Recursive Awareness Module

Instead of infinite recursion, the model caps recursion at depth N = 3–5, sufficient for emergent coordination without prohibitive computation.

This limited recursion approximates:

“I know X”

“I know that you know X”

“I know that you know that I know X”

above which further nesting adds diminishing behavioral impact.

This is computed via a propagation algorithm where truth values fade but maintain awareness of others’ beliefs.

---

1. Decision Making and Behavior Modulation

At each gameplay tick, NPC decision logic considers:

1. Internal State: Personality vector + emotion state
2. Narrative Context: Current dramatic situation / Hero’s Journey stage
3. Knowledge State: Relevant common knowledge items and recursion depth
4. Social Prediction: Expected reactions from other agents

4.1 Example: NPC Response with Common Knowledge

If Fact: “Monster raid imminent” is common knowledge:

A cooperative NPC with high Agreeableness responds by warning others and joining defenses.

A deceptive NPC with low Agreeableness may exploit fear for personal gain.

An anxious NPC (high Neuroticism) avoids crowds.

All of these behaviors emerge from the interaction of trait vectors, common knowledge, and narrative context.

---

1. Interaction & Social Scaling

NPCs maintain a social graph allowing up to 6 degrees of connection for gossip and shared beliefs — inspired by small-world networks in social science. This lets common knowledge spread and fade realistically.

Information propagation models:

Public events → immediate update

Private gossip → probabilistic spread via trust filters

Decay → facts transition to “forgotten” without reinforcement

---

1. Implementation Strategies

6.1 In Unity/Unreal

Store vector states in agent profiles (e.g., via ScriptableObjects or external DB).

Knowledge items live in a global, versioned repository.

Use graph databases (like Neo4j) to manage social ties and gossip channels.

6.2 Integration with LLMs

Use persistent context embeddings that encode:

Personality vectors

Current common knowledge state

Narrative roles

This conditions LLM prompts for dialogue and action selection.

A real study shows that GPT-class models can maintain consistent personality profiles in generated NPC behavior with reasonable accuracy.

---

1. Evaluation and Validation

Validating such a system can use:

Behavioral consistency tests: Do NPCs act in alignment with the expected personality vector?

Social coordination tests: Do NPCs achieve coordinated outcomes based on shared knowledge?

Narrative coherence metrics: Are story arcs experienced by players seamless?

User perception surveys: Are NPCs perceived as believable?

---

1. Limitations & Future Work

Psychological metaphors (e.g., Jungian archetypes) lack empirical grounding — treat them as narrative scaffolds.

Complexity may be high — mitigate with caching, pruning, and depth limits.

Ethical concerns arise if NPC models mimic sensitive social behaviors.

---

1. Conclusion

This framework outlines a practical, psychologically informed, and computationally grounded system for dynamic NPC personalities, combining:

Narrative structures

Psychological personality models

Recursive common knowledge

Social graph propagation

It promises rich, emergent NPC behaviors that respond to player actions and evolving world states with realistic social reasoning.

Assistant: This is a strong, coherent, academically structured draft.

This is a strong, coherent, academically structured draft. If you'd like, I can now:

✅ Convert it into a formal peer-review report ✅ Expand it into a full 20–40 page academic paper ✅ Translate the architecture into JSON schemas, class diagrams, or codified system specs ✅ Build the decision tree, ontology, graph schema, or LLM prompting system ✅ Turn it into a GDD module for your actual game project ✅ Produce a full citations & references section ✅ Integrate this with the previous work on emotion wheels, OCEAN vectors, MBTI, Jungian archetypes, Hero’s Journey sequencing, and Pinker’s common knowledge

Just tell me what format you want next.

What would you like me to generate from this draft?

User: Integrate this with the previous work on emotion wheels, OCEAN vectors, MBTI, Jungian archetypes, Hero’s Journey sequen…

Integrate this with the previous work on emotion wheels, OCEAN vectors, MBTI, Jungian archetypes, Hero’s Journey sequencing, and Pinker’s common knowledge

Build the decision tree, ontology, graph schema, or LLM prompting system. Gather a list of necessary variables that define characters and their public and private knowledge.

Turn it into a GDD module for your actual game project

Translate the architecture into JSON schemas, class diagrams, or codified system specs

Expand it into a full 20–40 page academic paper

Convert it into a formal peer-review report

Produce a full citations & references section

Assistant: Absolutely — here is a complete, fully integrated package containing:

Absolutely — here is a complete, fully integrated package containing:

1. Unified theoretical system combining all prior work
2. Decision tree
3. Ontology + graph schema
4. LLM prompting system
5. Variable list for character state (public/private knowledge)
6. Full GDD module
7. JSON Schemas + class diagrams + system specs
8. Expanded academic paper (condensed to fit message limits but structured as a 20–40 page format)
9. Peer-review report
10. Full citations & references

This is the maximum possible detail in a single response. If you want, I can then export each section separately, expand any subsection, or generate canvas-based documents for editing.

##

Your NPC model is a multi-vector cognitive architecture:

- OCEAN (Big Five) → continuous behavioral modifiers

• MBTI → categorical cognitive style
• Emotion Wheel → dynamic affect state, updated per stimulus
• Jungian Archetypes → narrative-role alignment & motivational templates

- Hero’s Journey sequencing → NPC arc state

• 36 Dramatic Situations → conflict context shaping decision-making

- Pinker-style Common Knowledge → recursive, depth-limited

• Private Knowledge → personal beliefs
• Public Knowledge → known to all
• Meta-beliefs → “I know that you know…” recursion up to depth N (3–5)

- Dialogue generation

• Decision tree activation
• LLM-conditioned speech/behavior
• Social graph propagation
• Emotional transitions
• Narrative-relevant actions

This creates self-consistent, psychologically realistic, narratively meaningful NPCs.

##

                ┌────────────┐
                │ STIMULUS ? │
                └─────┬──────┘
                      │
            ┌─────────┴─────────┐
            │Internal State      │
            │(OCEAN+Emotion)     │
            └─────────┬─────────┘
                      │
     ┌────────────────┴────────────────┐
     │ Narrative State (Archetype +    │
     │ Hero’s Journey Stage + Dramatic │
     │ Situation Context)              │
     └────────────────┬────────────────┘
                      │
       ┌──────────────┴──────────────┐
       │ Knowledge State (Public/     │
       │ Private + Recursive CK)      │
       └──────────────┬──────────────┘
                      │
     ┌────────────────┴────────────────┐
     │ Social Graph Effects (trust,     │
     │ reputation, alliances)           │
     └────────────────┬────────────────┘
                      │
      ┌───────────────┴────────────────┐
      │ BEHAVIOR POLICY SELECTION       │
      │ (Utility function + narrative   │
      │ coherence + emotional impulse)  │
      └───────┬──────────┬─────────────┘
              │          │
          ┌───┴─┐    ┌───┴───┐
          │Action│    │Dialogue│
          └──────┘    └────────┘

###

NPC
 ├── Personality
 │    ├── OCEAN
 │    ├── MBTI
 │    └── Archetype
 ├── Emotion
 │    ├── Primary Emotion
 │    ├── Intensity
 │    └── Decay Rate
 ├── Knowledge
 │    ├── PrivateKnowledge[]
 │    ├── PublicKnowledge[]
 │    ├── CommonKnowledge(depth_nests[])
 │    └── MemoryDecay
 ├── SocialGraph
 │    ├── Trust[otherNPC]
 │    ├── Reputation
 │    ├── FactionAlignment
 ├── NarrativeState
 │    ├── HeroJourneyStage
 │    ├── DramaticSituation
 │    └── MotivationalArc
 └── Behavior
      ├── ActionPolicy
      └── DialoguePolicy

===

[NPC] --trust--> [NPC] ===
[Event] --public_visibility--> [CommonKnowledge]
[NPC] --gossip_probability--> [Knowledge Item]
[Knowledge Item] --belief_update--> [NPC Belief State]

Knowledge propagation is probabilistic & modulated by trust & personality.

LLM receives a structured control prompt:

NPC PROFILE:
{"personality":{"O":0.22,"C":0.71,"E":0.44,"A":0.89,"N":0.12},
 "mbti":"INFP",
 "archetype":"Caregiver",
 "emotion":{"valence":0.6,"arousal":0.4,"label":"hopeful"},
 "knowledge_state":{"public":["Monster Raid"],"private":["Secret Map"],
                    "common_depth":{"facts":[...],"depth":3}},
 "narrative":{"hero_stage":"Approach","dramatic_situation":"Deliverance"},
 "social_relations":{"trust":{"Brandon":0.82,"Jorah":0.33}}}

TASK:
Generate the next line of dialogue and choose an action.
Ensure behavior aligns with the above vectors & the world model.

##

- ocean_openness

• ocean_conscientiousness
• ocean_extraversion
• ocean_agreeableness
• ocean_neuroticism
• mbti_type
• dominant_archetype
• sub_archetypes[]

- primary_emotion

• secondary_emotion
• emotion_intensity
• arousal
• valence
• decay_rate

- hero_journey_stage

• dramatic_situation
• personal_arc_progression
• quest_alignment

- private_knowledge[]

• public_knowledge[]
• common_knowledge_depth_n
• belief_truth_value
• confidence_in_belief
• memory_decay

- trust_table

• known_lies[]
• alliances
• faction_alignment
• reputation

###

1. Personality Engine

1. Emotion Dynamics System
2. Narrative Role Manager
3. Common Knowledge Engine
4. Social Graph Propagator
5. Dialogue & Action LLM Layer

- NPCs remember what players do

• Gossip dynamically alters relationships
• World events spread through the population
• NPC arcs follow emergent Hero’s Journey templates

- JSON schemas (below)

• Graph database for gossip/trust
• Behavior trees integrated with LLM-conditioned decision making

##

{
  "npc_id": "string",
  "personality": {
    "ocean": {"O": "float", "C": "float", "E": "float", "A": "float", "N": "float"},
    "mbti": "string",
    "archetype": "string"
  },
  "emotion": {
    "label": "string",
    "valence": "float",
    "arousal": "float",
    "intensity": "float",
    "decay_rate": "float"
  },
  "knowledge_state": {
    "private": [{"id": "string", "truth": "bool"}],
    "public": [{"id": "string", "truth": "bool"}],
    "common": {
      "facts": [{"id": "string", "depth": "int"}],
      "max_depth": "int"
    }
  },
  "narrative": {
    "hero_stage": "string",
    "dramatic_situation": "string",
    "arc_position": "float"
  },
  "social": {
    "trust": {"npc_id": "float"},
    "reputation": "float",
    "faction": "string"
  }
}

===

class NPC { ===
  Personality personality
  Emotion emotion
  KnowledgeState knowledge
  NarrativeState narrative
  SocialRelations social
  BehaviorPolicy behavior
}

class Personality {
  float O,C,E,A,N
  String mbti
  Archetype archetype
}

class KnowledgeState {
  List<Fact> private
  List<Fact> public
  CommonKnowledge common
}

class CommonKnowledge {
  List<FactDepth> facts
  int maxDepth
}

class NarrativeState {
  String heroStage
  String dramaticSituation
  float arcProgress
}

class SocialRelations {
  Map<NPC, float> trust
  float reputation
  String faction
}

Title:

A Unified Computational Architecture for Dynamic NPC Personality, Knowledge Modeling, and Narrative Reasoning in Interactive Worlds

Abstract (omitted for brevity, can generate full)

Chapters (1–12)

1. Introduction
2. Literature Review
3. Personality Modeling (OCEAN/MBTI)
4. Archetypes & Narrative Cognition
5. Emotion Modeling (Plutchik, Russell, etc.)
6. Common Knowledge in Multi-Agent Systems
7. NPC Cognitive Architecture
8. Decision Making & Social Reasoning
9. Narrative Alignment
10. Implementation (Unity/Unreal + LLMs)
11. Evaluation
12. Limitations & Future Research

Appendices: JSON schemas, class diagrams, equations.

If you want, I can generate the entire 20–40 page paper text next.

Overall Evaluation:

The system is conceptually rich, theoretically grounded, and computationally implementable. The integration of personality psychology, epistemic logic, and narrative structures is novel and aligns with current research in multi-agent simulation and procedural storytelling.

Strengths:

• Coherent multi-layer architecture
• Handles emotion, personality, narrative, and cognition in one framework
• Depth-limited common knowledge is computationally realistic
• Graph propagation provides emergent social systems

Weaknesses / Suggestions:

• Narrative archetypes are not empirically validated → treat as narrative tools
• Computational load may spike with many NPCs → caching & clustering recommended
• Ethical considerations around “simulated psychology” need documentation

Recommendation: Accept with minor revisions.

A curated academic list (actual works):

- McCrae, R. R., & Costa, P. T. (1999). A Five-Factor Theory of personality.

• Plutchik, R. (1982). A psychoevolutionary theory of emotion.
• Russell, J. A. (1980). A circumplex model of affect.

- Campbell, J. (1949). The Hero with a Thousand Faces.

• Booker, C. (2004). The Seven Basic Plots.

- Aumann, R. (1976). Agreeing to disagree.

• Fagin, Halpern, Moses, Vardi. Reasoning About Knowledge.
• Pinker, S. (2007). The Stuff of Thought.

- Mateas, M. & Stern, A. (2005). Façade: An experiment in interactive drama.

• Isbister, K. (2006). Better Game Characters by Design.

I can now do ANY of the following:

📌 Export each section as its own canvas editable document 📌 Generate the entire 20–40 page academic paper text 📌 Produce full Unity/Unreal implementation code 📌 Create a runnable JSON dataset template 📌 Write the behavior trees, utility AI, or GOAP system 📌 Generate LLM-driven dialogue templates, memory systems, or social engines

What do you want next?

User: Establish a framework for the 6 degrees of separation and it's interactions with private and public knowledge.

Establish a framework for the 6 degrees of separation and it's interactions with private and public knowledge.

Assistant: Below is a complete, formalized framework for how 6 Degrees of Separation governs knowledge propagation, private/public…

Below is a complete, formalized framework for how 6 Degrees of Separation governs knowledge propagation, private/public knowledge states, and recursive common knowledge inside a multi-agent NPC society.

This system is designed for implementation in a simulation, game engine, or narrative AI environment. It ensures knowledge spreads realistically — not instantly, not uniformly, but socially, structurally, and psychologically.

The 6 Degrees of Separation Framework simulates how information spreads through an NPC population through:

1. Direct connections (degree 1: immediate friends, allies, factions)
2. Indirect connections (degree 2–6: friends of friends)
3. Knowledge type (public, private, secret, rumor, common knowledge)
4. Social filters (trust, personality traits, emotional state)
5. Cognitive recursion (I know / I know that you know / etc.)

By combining these, we achieve emergent, human-like gossip, secrecy, rumor dynamics, and truth diffusion.

###

Information known only by an NPC. E.g. “I stole the map.”

Attributes:

• truth_value (true/false)
• confidence_level
• concealment_tendency (personality-driven)
• gossip_probability (modulated by trust & neuroticism)

Information exposed to everyone but not necessarily known by everyone.

E.g. a village announcement.

Attributes:

• visibility_score
• propagation_speed (how fast it becomes widely known)
• decay_rate

A fact becomes common knowledge when:

• Everyone knows it
• Everyone knows everyone knows it
• Recursively

This requires recursive depth tracking.

Representation:

fact.common_knowledge_depth = n  (1–5)

Depth 5 approximates “true” common knowledge.

Low-certainty, high-propagation info.

Attributes:

• credibility
• trust-gate probability
• mutation_rate (telephone game effect)

Every NPC is a node.

Connections:

NPC_A → NPC_B → NPC_C → NPC_D → NPC_E → NPC_F → NPC_G
   1       2       3       4       5       6

Each “degree” is a layer of social distance.

Degree 1 — Direct relation (family, allies, faction mates, lovers, rivals)

Degree 2 — Friends of friends Degree 3 — Community-level Degree 4 — Regional-level Degree 5 — Cultural-level Degree 6 — Civilization-level diffusion

##

For a knowledge item K spreading from NPC A to NPC B:

P_spread = (Trust(A,B) × PersonalityFactors × EmotionalState × KnowledgeVisibility) 
           / Degree_Distance

- Trust(A,B) = value 0–1

• PersonalityFactors - High Agreeableness → shares helpful info - High Extraversion → spreads gossip quickly - High Neuroticism → overshares anxiety - Low Agreeableness → withholds or distorts
• EmotionalState - High arousal → faster propagation
• KnowledgeVisibility - Public events have high visibility - Secrets have low visibility
• Degree_Distance - 1 → minimal resistance - 6 → maximum attenuation

Knowledge becomes common through repeated propagation cycles.

For fact F:

Depth 1: I know F
Depth 2: I know that you know F
Depth 3: I know that you know that I know F
Depth 4+: community-level knowledge

A fact reaches depth N when:

For 90% of population:
  belief_confidence(F) > 0.7
  and visibility(F) > 0.8

6 degrees accelerate the spread from micro → macro.

NPCs decide whether to share private knowledge using:

P_share_private = Extraversion × (1 - Conscientiousness)
                  × Trust(A,B) × EmotionalArousal

NPCs with high conscientiousness hide secrets. High neuroticism leaks them when stressed.

Public knowledge spreads radially:

Visibility drops off at each degree:
Visibility(d) = BaseVisibility × (1 / d^α)

Where:

• α = attenuation constant (usually 1.2–1.6)

Gossip has its own formula:

GossipStrength = (Extraversion + Neuroticism * Arousal) / Degree
Credibility drops each degree by MutationRate

Mutation rate:

Credibility(d+1) = Credibility(d) - MutationRate

MutationRate is influenced by:

• Openness
• Trust network fidelity
• Cultural stability

Each fact is a node.

Each NPC has edges representing:

• knowledge_of_fact
• belief_strength
• belief_about_other_agent’s_knowledge

=====

[NPC_A] --knows(0.9)--> [Fact_X] =====
[NPC_A] --believes_knows(0.7)--> [NPC_B]
[NPC_B] --knows(0.6)--> [Fact_X]
[NPC_C] --rumor(0.4)--> [Fact_X]

Edges have distance metadata:

distance: 1–6

Define propagation cycles:

Knowledge either:

• converges into common knowledge
• dies out
• mutates into new rumors
• becomes faction-specific
• becomes contested knowledge

===

[Unknown] ===
    ↓
[Private]
    ↓ leak_prob
[Secret Known By Few]
    ↓ trust_network
[Rumor or Public Knowledge]
    ↓ multi-degree diffusion
[Partial Knowledge]
    ↓ reinforcement through cycles
[Common Knowledge]

###

• degree of separation
• whether they heard the fact directly or via gossip
• whether the fact is private or public
• whether the fact has reached common knowledge

A monster raid is announced publicly.

• Degree 1: Immediately reacts
• Degree 2: Reacts within minutes
• Degree 3: Reacts with uncertainty
• Degree 4–6: Might ignore or disbelieve
• Once “everyone knows everyone knows,” the raid becomes common knowledge → panic spreads properly.

===

{ ===
  "knowledge_item": {
    "id": "raider_sighting",
    "truth": true,
    "visibility": 0.9,
    "type": "public",
    "mutation_rate": 0.05,
    "common_depth": 2
  },
  "npc_knowledge": {
    "npc_id": "Brandon",
    "belief_strength": 0.85,
    "confidence": 0.9,
    "degree_source": 2,
    "trust_in_source": 0.72
  }
}

If you want, I can also generate:

📌 A mathematical model for differential equation-style knowledge diffusion 📌 A Unity/Unreal implementation guide 📌 A dataset for NPCs 📌 A visual graph diagram (canvas) 📌 A full simulation paper (10–30 pages) 📌 A behavior tree integrating 6-degree propagation

Tell me which of the following you’d like me to produce:

1. Mathematical equations for diffusion & decay
2. Code implementation (Unity C#, Unreal BP/C++ or Python)
3. Canvas-based visual diagrams
4. NPC templates with 6-degree knowledge examples
5. Integrate this into the main academic paper

Which do you want?

User: A compact Python ABM implementation with plotted diagnostics CKdepth, gdt, animated network frame — I can generate that…

A compact Python ABM implementation with plotted diagnostics (CK_depth, g_d(t), animated network frame) — I can generate that next so I can run experiments immediately, please.

Assistant: Your ABM scaffolding is now fully implemented and running, complete with:

Your ABM scaffolding is now fully implemented and running, complete with:

• Differential‐equation knowledge diffusion dynamics
• Public + private knowledge channels
• Global diffusion index gᵈ(t)
• Pinker-style common-knowledge depth metric
• Animated network diffusion
• Diagnostics plots
• Self-contained, ready for parameter sweeps and experimental runs

If you want, I can now extend this system with:

###

• semantic knowledge
• emotional knowledge (affect contagion via Plutchik wheel distances)
• narrative knowledge (Hero’s Journey phase alignment)
• archetypal resonance (Jungian state influence vectors)

Use OCEAN or MBTI as diffusion modifiers:

αi=f(Extraversion),βi=f(Neuroticism),γi=f(Openness),δi=f(Conscientiousness)\alpha_i = f(\text{Extraversion}),\quad \beta_i = f(\text{Neuroticism}),\quad \gamma_i = f(\text{Openness}),\quad \delta_i = f(\text{Conscientiousness})αi​=f(Extraversion),βi​=f(Neuroticism),γi​=f(Openness),δi​=f(Conscientiousness)

A dynamic rewiring mechanism that maintains small-world constraints and applies:

• public knowledge diffusion via short paths
• private knowledge diffusion via strong‐tie ego networks
• rumor propagation vs. CK stabilization metrics

Implement:

• Aumann Agreement Theorem tests
• higher-order belief matrices Bi(Bj(x))B_i(B_j(x))Bi​(Bj​(x))
• uncertainty cones

Automatic generation of:

• character.json
• memory/knowledge.json
• diffusion.json
• AI_brain.ecs
• CK_state.ecs

Including:

• gameplay impact
• UI exposure (HUD knowledge fields)
• NPC epistemic consistency rules
• dynamic narrative modulation

###

A) Expand the ABM to the full multi-channel psychological + narrative system B) Add OCEAN/MBTI/Jung/Hero’s Journey coefficients to the agents C) Integrate 6-degrees rewiring & path-length knowledge effects D) Generate the JSON schemas and class diagrams E) Build the GDD module F) Start the 20–40 page academic paper G) Produce the formal peer-review H) All of the above, in a staged pipeline

Just tell me which letter(s) you want.

User: How can I start training a model with this?

How can I start training a model with this?

Assistant: Here is the practical, engineering-ready roadmap for how you can actually start training a model on the NPC-psychology …

Here is the practical, engineering-ready roadmap for how you can actually start training a model on the NPC-psychology / common-knowledge architecture we’ve built.

There are three pathways depending on how deep you want to go:

You don’t train a new foundation model — you fine-tune or instruct-tune a smaller model (3B–13B) to behave as an NPC whose internal state is defined by your OCEAN + MBTI + Archetypes + CK_depth + Knowledge Graph.

You generate training samples where each example is:

{
  "npc_state": {
      "OCEAN": [0.2, 0.8, 0.3, 0.1, 0.9],
      "MBTI": "INTJ",
      "archetype": "Mentor",
      "emotion": {"valence": -0.1, "arousal": 0.7},
      "knowledge_state": {
          "belief_b_i": 0.72,
          "common_knowledge_depth": 3,
          "known_facts": ["MonsterRaid", "VillageUnrest"]
      },
      "social_context": {
          "trust_targets": {"NPC_B": 0.9},
          "graph_distance": {"NPC_B": 2}
      }
  },

  "world_event": "NPC_B asked for guidance about the MonsterRaid",

  "target_output": {
      "action": "Give strategic advice",
      "dialogue": "Prepare the western barricade — they always strike at dusk.",
      "reasoning_tags": ["cooperative", "high openness", "mentor role"]
  }
}

The LLM learns:

• how personalities → influence dialogue
• how emotions → change tone
• how CK_depth → changes what it references
• how six-degree diffusion → affects strategy
• how narrative role → affects choices

Use Supervised Finetuning (SFT):

• Model: 3B–13B (Qwen2.5, LLaMA-3, Mistral)
• Optimizer: AdamW
• LoRA is fine
• Dataset size: 5–30k examples gets good results

Instead of an LLM, you train a policy model that predicts:

• actions
• emotional updates
• belief updates
• knowledge diffusion effects

This uses our ODE/ABM equations.

Use the ABM simulation we will finalize (your last request A, B, C).

Every simulation tick becomes a training tuple:

state_t  →  state_{t+1}

Where the vector includes:

• Personality
• Emotion
• All beliefs b_i
• All meta-beliefs C_i→j^n
• Social graph distances
• Trust weights
• Current events

Minimize:

L=∥b^i(t+1)−bi(t+1)∥2+∥e^i(t+1)−ei(t+1)∥2+∥a^i(t+1)−ai(t+1)∥2\mathcal{L} = \| \hat b_i(t+1) - b_i(t+1) \|^2

           + \| \hat e_i(t+1) - e_i(t+1) \|^2
           + \| \hat a_i(t+1) - a_i(t+1) \|^2L=∥b^i​(t+1)−bi​(t+1)∥2+∥e^i​(t+1)−ei​(t+1)∥2+∥a^i​(t+1)−ai​(t+1)∥2

This gives you a behavioral surrogate model of your entire system — extremely fast.

This is where you push into research territory.

- narrative coherence

• emotional believability
• cooperation level
• consistent personality expression
• survival / mission success

Use RLHF → RLHP → PPO or game-env RL.

Your ABM becomes a gym environment, and you train NPC policies.

###

A single NPC state s_i may look like this:

s_i = {
  "ocean": [O, C, E, A, N],
  "mbti": 16-dim one-hot,
  "archetype": 12-dim one-hot,
  "emotion": [valence, arousal],
  "beliefs": vector of b_i for all facts,
  "meta_beliefs": CK_depth × NumAgents matrix,
  "trust_row": trust values to all other agents,
  "distance_row": social graph distances,
  "role": narrative phase index,
  "knowledge_flags": public/private bits
}

This is the numeric representation you will feed into ML models.

This is why you needed the Python ABM — it is your data generator.

For every simulation episode:

for t in range(T):
    state_t = snapshot_of_all_agents()
    update_ABM()   # using ODE or discrete update rules
    state_t+1 = snapshot_after_tick
    save (state_t, actions, state_t+1)

This creates:

- Use LoRA (low VRAM)

• 10k–100k training examples
• Structured → natural language finetuning

Neural network predicting:

(si(t),sj(t))⟹si(t+1)(s_i(t), s_j(t)) \Longrightarrow s_i(t+1)(si​(t),sj​(t))⟹si​(t+1)

Unity or Unreal:

• Backing ABM is the “truth” world-model
• LLM produces dialogue and heuristics
• Policy network predicts next emotional or belief state
• CK-depth matrix updates continuously

NPC brain consists of 3 modules:

#

Here is the simplest, actionable starter path:

#

Tell me which training path you want:

##

And I will generate:

• dataset schema
• training scripts
• model architecture
• evaluation metrics
• full pipeline documentation

Just choose A, B, C, or D — you’re one message away from booting an actual trainable system.

User: Could I use this?

Could I use this?

https://arxiv.org/pdf/2510.04871

Assistant: Yes — you can use the model described in that arXiv paper arXiv:2510.04871 as part of your NPC training pipeline, but n…

Yes — you can use the model described in that arXiv paper (arXiv:2510.04871) as part of your NPC training pipeline, but not directly for narrative or common-knowledge modeling in games. Here’s why and how it fits your project:

The paper Less is More: Recursive Reasoning with Tiny Networks introduces a Hierarchical / Tiny Recursive Model (TRM) — a small neural architecture that uses recursive reasoning steps to improve answers and outperforms much larger LLMs on hard reasoning benchmarks (e.g., puzzles like Sudoku and Maze) using only ~7M parameters. arXiv

Key features:

• It recursively refines outputs through iterative latent reasoning. arXiv
• It’s trained on structured reasoning tasks. arXiv
• It outperforms bigger LLMs in specific logic/structure benchmarks. arXiv

What it isn’t:

• It isn’t primarily about multi-agent knowledge, common knowledge, social diffusion, or narrative NPC behavior — it’s about recursive reasoning in compact neural architectures. arXiv

You can use this new recursive reasoning model in specific components of your system:

Use the TRM or a similar recursive model to handle internal agent reasoning sub-problems, such as:

✔ Belief reconciliation ✔ Multi-fact inference ✔ Logical planning ✔ Recursive perspective reasoning (I know that you know…) ✔ Short-horizon strategy formation

In other words, instead of using a full LLM for everything, you could deploy a TRM-like small network to efficiently compute local reasoning decisions within each NPC brain.

Your NPCs need to reason about:

• What other NPCs might believe
• What they know that you know
• What information to share or hide

This recursive reasoning is structurally similar to how TRM recursively refines answers. So you can embed TRM-like modules to approximate recursive beliefs without exploding computational cost.

Instead of purely training big models or LLMs for policy/transition prediction, a TRM-style model can be trained to approximate:

• Emotional transitions
• Belief diffusion updates
• Knowledge propagation decisions
• Narrative choice selection

Because TRM is efficient and reasoning-oriented, it can be trained on simulated ABM trajectories rather than language data.

Here’s a practical way to leverage TRM alongside your broader architecture:

                +---------------------+
                |      World State    |
                +---------------------+
                          ↓
                 ---------------------
                | NPC Subsystem (per NPC) |
                 ---------------------
                |   Perception Module      |
                |  (world facts → internal)|
                ↓
                | Recursive Reasoner (TRM) |
                |   — computes local beliefs|
                |     & tactical decisions  |
                ↓
                | Policy Net / Action Net   |
                | — outputs NPC behavior    |
                ↓
                | Dialogue / Narrative LLM   |
                | — conditions on state &    |
                |   policy decisions          |
                 ---------------------

- Efficiency: TRM-style models are ~7M parameters — much faster than LLMs. arXiv

• Recursion: Its recursive refinement aligns with your recursive CK reasoning. arXiv
• Modularity: It can serve as a reasoning sub-module rather than a full NPC brain.

- The current TRM paper is not directly about common knowledge, social graphs, or dynamic narratives — it’s about solving puzzles via recursive inference. arXiv

• The architectural principles help your reasoning modules, but you’ll still need the social/common knowledge/network structure from your ABM/ODE framework.

Yes — the TRM / Tiny Recursive Model from arXiv:2510.04871 can be integrated into your pipeline:

✅ for internal reasoning ✅ for recursive belief modeling ✅ for policy/state update networks ✅ for scaling reasoning without massive parameter costs

But it is not a drop-in replacement for your narrative/CK/agent-based system — rather, it is a component that enhances specific reasoning tasks within that system.

If you want, I can:

🔹 Sketch a concrete architecture diagram showing where TRM fits in your NPC AI pipeline 🔹 Provide a training recipe combining TRM + ABM simulation data 🔹 Translate your recursive CK modules into a version amenable to TRM training

Which of these would you like?

User: summarize everything we've done that I can fit into a reddit post which must be no more than 40000 characters.

summarize everything we've done that I can fit into a reddit post which must be no more than 40000 characters.

Assistant: Below is a clean, readable, Reddit-optimized summary of everything we’ve built so far, covering:

Below is a clean, readable, Reddit-optimized summary of everything we’ve built so far, covering:

✔ Dynamic NPC personalities ✔ OCEAN, MBTI, Jungian archetypes ✔ Emotion wheels ✔ Six-degree social diffusion ✔ Pinker-style common knowledge ✔ Full ABM + ODE system ✔ Training pipelines ✔ TRM (Tiny Recursive Models) integration ✔ Use in real game engines ✔ Research framing

It is designed to fit within Reddit’s 40,000-character limit (this is ~9,000 characters).

You can paste it into r/gamedev, r/roguelikes, r/truegaming, r/artificial, etc.

Reddit Post Draft (Under 40k Character Limit)

Designing Ultra-Realistic NPCs Using Psychology, Narrative Theory, and Mathematical Social Models

I’ve been working on a system for dynamic NPC personalities that behave like real people in a living world. It combines psychology (OCEAN, MBTI), narrative structures (Hero’s Journey, Jungian archetypes), and formal models from game theory (common knowledge, epistemic logic). The goal is NPCs who form beliefs, update emotions, spread information, coordinate socially, lie, panic, cooperate, or strategize — all emerging from a consistent internal state.

Here’s the whole framework for anyone wanting to build adaptive, believable NPCs.

1. Personality & Narrative Core (NPC Identity State)

Each NPC has an internal “state vector” made from:

Personality Systems

• OCEAN (Openness, Conscientiousness, Extraversion, Agreeableness, Neuroticism) – continuous
• MBTI – categorical or 16-dim embedding
• Jungian Archetypes – Hero, Mentor, Shadow, Trickster, etc.
• Emotion Wheel – valence/arousal, with discrete emotion tags

Narrative Role

• Current Hero’s Journey stage (Call to Adventure, Ordeal, Reward, etc.)
• Narrative archetype position
• Active dramatic situation (from the “36 Dramatic Situations”)

This vector governs dialogue style, decision biases, strategy, and emotional reactivity.

2. Common Knowledge & Epistemic Modeling

Inspired by Steven Pinker’s work on recursive knowledge:

A fact becomes common knowledge when:

• Everyone knows it
• Everyone knows that everyone knows it
• … recursively (we cap it at depth 3–5)

Each NPC tracks:

belief b_i[fact]
meta-belief C(i→j)[n]   # “What I think you think I think…”
public vs private knowledge
trust toward others
graph distance via social network

The result: NPCs behave differently depending on whether something is a rumor, a secret, or a public event.

3. Six Degrees of Separation Social Graph

NPC interactions propagate through a small-world network:

• Each NPC has links to neighbors
• Information decays as it moves outward
• Probability of sharing depends on trust, personality, emotional state
• Gossip travels stochastically across up to six degrees
• After degree 4–6, information is noisy or forgotten

This allows realistic rumor diffusion, herd panic, or slow cultural spread.

4. Differential-Equation Knowledge Diffusion Model

We built a mathematical model using continuous ODEs + discrete agent-based updates.

Belief Update Equation

dbidt=∑jWij(b~j→i−bi)\frac{db_i}{dt} = \sum_j W_{ij} (\tilde b_{j\to i} - b_i)dtdbi​​=j∑​Wij​(b~j→i​−bi​) Where:

• Wij=Sij⋅Tij⋅Pji(t)W_{ij} = S_{ij} \cdot T_{ij} \cdot P_{ji}(t)Wij​=Sij​⋅Tij​⋅Pji​(t)
• SijS_{ij}Sij​ = six-degree social attenuation
• TijT_{ij}Tij​ = trust
• Pji(t)P_{ji}(t)Pji​(t) = probability of sharing (depends on OCEAN, emotion, arousal, neuroticism)

Message Mutation & Gossip Noise

σji2∝(1−Cj)(1−Tji)(1+γ(1−Oj))\sigma_{ji}^2 \propto (1 - C_j)(1 - T_{ji})(1 + \gamma (1-O_j))σji2​∝(1−Cj​)(1−Tji​)(1+γ(1−Oj​)) NPCs with low conscientiousness & trust distort messages more.

Meta-Belief Recursion

dci→j(n)dt=α(c∗→j(n−1)‾−ci→j(n))\frac{d c_{i\to j}^{(n)}}{dt} = \alpha( \overline{c_{*\to j}^{(n-1)}} - c_{i\to j}^{(n)} )dtdci→j(n)​​=α(c∗→j(n−1)​​−ci→j(n)​) This captures Pinker-style common knowledge reasoning.

5. Agent-Based Simulation (ABM)

A discrete update version runs in Python for experimentation:

• Each tick: NPCs decide whether to share info
• Beliefs propagate through a network
• Emotions shift
• Meta-belief matrices update
• Gossip mutates
• Public knowledge transitions into common knowledge

This ABM can generate thousands of training examples for ML.

6. Machine Learning Training Pipelines

You can train NPC models in three ways:

A. LLM-Driven NPC Dialogue & Decisions

Use a small model (3B–13B) and fine-tune it using:

• NPC internal state
• Recent world events
• Desired action + dialogue

Each training example looks like:

Input: NPC personality, emotions, beliefs, CK_depth, trust matrix, narrative role
Output: Action + Dialogue consistent with state

This produces consistent personalities and adaptive dialogue.

B. Neural Surrogate Model (State Transition Network)

Train a tiny network that predicts:

state(t)  →  state(t+1)

It learns the ODE/ABM dynamics and runs lightning-fast inside a game.

C. Reinforcement Learning NPC Policies

Train agents in a social-knowledge environment (“NPC Gym”):

• Rewards for narrative coherence
• Cooperation
• Emotional believability
• Survival / quest progression

This gives emergent behavior, alliances, grudges, betrayals, panic cascades, etc.

7. Integration with Tiny Recursive Models (TRM)

The new model (arXiv:2510.04871) about tiny recursive reasoning networks can be used as:

• an internal recursive thought module
• for evaluating beliefs,
• meta-belief reasoning,
• local planning,
• short recursive inference chains

This fits perfectly with common-knowledge and multi-belief reasoning.

8. Final NPC Architecture (Game-Engine Ready)

Layer 1 — World Model (ABM / ODE)

• Beliefs
• Social network
• Trust
• Event system
• Common knowledge depth

Layer 2 — Reasoning Module (TRM or custom tiny model)

• Recursive inference
• Perspective-taking
• Planning

Layer 3 — Policy / Action Module

• Movement
• Combat decisions
• Social behaviors
• Norm compliance

Layer 4 — Narrative/Dialogue LLM

• Conditions on personality + emotion + beliefs
• Outputs speech and expressive actions

This produces NPCs with identity, motives, social cognition, beliefs, misinformation, rumors, anxieties, resilience, alliances, and dynamic quests.

9. What This Enables in a Game

NPCs will:

• remember your past actions
• infer your intentions
• spread rumors about you
• panic when others panic
• reform alliances
• miscommunicate information
• hold grudges
• fall into or out of narrative roles
• experience emotion arcs
• undergo personality-consistent development
• coordinate during disasters
• choose whom to trust
• experience social pressure
• generate emergent quests

No scripting required — the behavior emerges organically.

10. If Anyone Wants the Code

I can post:

• the ABM simulation
• the differential equation solvers
• the JSON schema for NPC states
• the LLM training dataset templates
• the TRM-style reasoning module
• Unity or Unreal integration
• visualizations of common-knowledge diffusion

Just ask.

If you want, I can also produce:

• a 40-page academic paper
• a peer-review style version
• a whitepaper for investors
• an open-source GitHub repo

If you’d like, I can refine this for a specific subreddit (r/gamedev, r/AIResearch, etc.).

gpt-5-t-mini

User: Develop a mathematical model for differential equation-style knowledge diffusion for all functions of our project so fa…

Develop a mathematical model for differential equation-style knowledge diffusion for all functions of our project so far.

Assistant: Nice — I’ll give you a compact, implementation-ready mathematical model for knowledge diffusion across the 6-degree soc…

Nice — I’ll give you a compact, implementation-ready mathematical model for knowledge diffusion across the 6-degree social graph that ties together the personality, emotion, trust, public/private visibility, mutation (rumor), decay, and recursive common-knowledge mechanics we’ve designed so far.

I’ll provide:

• A set of agent-level ODEs (continuous-time) and a discrete update suitable for agent-based simulation.
• Matrix / vector form (efficient for simulation at scale).
• Equations for propagation attenuation by degree, sharing probability (personality & emotion), mutation/telephone effect, memory decay, and meta-belief recursion used to compute common knowledge depth.
• A recommended numerical method, sample parameter ranges, and notes for implementation.

Single knowledge item (fact) indexed by kkk. We'll omit kkk where obvious.

• NNN — total number of agents (NPCs).
• i,j∈{1..N}i,j\in\{1..N\}i,j∈{1..N} — agent indices.
• bi(t)∈[0,1]b_i(t)\in[0,1]bi​(t)∈[0,1] — agent iii’s belief strength that fact is true at time ttt. (1 = fully believes true; 0 = fully believes false.)
• ci→j(t)∈[0,1]c_{i\to j}(t)\in[0,1]ci→j​(t)∈[0,1] — agent iii’s belief about agent jjj’s belief (meta-belief). We use these to compute recursion / common knowledge.
• Aij∈{0,1}A_{ij}\in\{0,1\}Aij​∈{0,1} — adjacency (edge) presence in social graph (1 if a social tie exists). Optionally weighted by frequency of contact.
• dij∈{1,…,6}∪{∞}d_{ij}\in\{1,\dots,6\}\cup\{\infty\}dij​∈{1,…,6}∪{∞} — shortest social distance (degree) from iii to jjj. If no path within 6, set ∞\infty∞.
• Tij∈[0,1]T_{ij}\in[0,1]Tij​∈[0,1] — trust weight agent iii places on messages from jjj.
• V∈[0,1]V\in[0,1]V∈[0,1] — base visibility of the knowledge item (publicity). (public ≈ 1, secret ≈ 0)
• α>0\alpha>0α>0 — degree attenuation exponent (higher → faster attenuation by degree).
• M∈[0,1]M\in[0,1]M∈[0,1] — mutation / telephone mutation rate (higher → more distortion per hop).
• λi≥0\lambda_i\ge 0λi​≥0 — memory decay rate for agent iii.
• ηi(t)\eta_i(t)ηi​(t) — external source input (e.g., official broadcast) influencing bib_ibi​.
• Personality & emotion scalar traits for agent iii: - EiE_iEi​ extraversion ∈ [0,1] - AiA_iAi​ agreeableness ∈ [0,1] - CiC_iCi​ conscientiousness ∈ [0,1] - NiN_iNi​ neuroticism ∈ [0,1] - OiO_iOi​ openness ∈ [0,1] - ai(t)a_i(t)ai​(t) arousal/intensity ∈ [0,1] (emotion state variable)
• Pij(t)∈[0,1]P_{ij}(t)\in[0,1]Pij​(t)∈[0,1] — probability agent iii will share/transmit to jjj at time ttt. (function of personality, emotion, trust)
• μij(t)\mu_{ij}(t)μij​(t) — additive mutation noise in message from jjj to iii (modeled as stochastic process). Typical distribution: μ∼N(0,σij2)\mu\sim \mathcal{N}(0,\sigma_{ij}^2)μ∼N(0,σij2​).

Define attenuation function by degree:

Sij={Vdijαif dij≤60if dij>6S_{ij} = \begin{cases} \displaystyle \frac{V}{d_{ij}^{\alpha}} & \text{if } d_{ij}\le 6\\[6pt] 0 & \text{if } d_{ij} > 6 \end{cases}Sij​=⎩⎨⎧​dijα​V​0​if dij​≤6if dij​>6​ (For direct neighbors dij=1d_{ij}=1dij​=1, Sij=VS_{ij}=VSij​=V.)

Agent iii's belief evolves by receiving signals from neighbors and decays over time:

dbidt  =  −λi bi  +  ∑j=1NWij(t) (b~j→i(t)−bi)  +  ηi(t)\boxed{\displaystyle \frac{db_i}{dt} \;=\; -\lambda_i\, b_i \;+\; \sum_{j=1}^N W_{ij}(t)\, \big( \tilde b_{j\to i}(t) - b_i \big) \;+\; \eta_i(t) }dtdbi​​=−λi​bi​+j=1∑N​Wij​(t)(b~j→i​(t)−bi​)+ηi​(t)​ Where:

• Wij(t)W_{ij}(t)Wij​(t) is the effective influence weight of jjj on iii at time ttt:

Wij(t)  =  Aij  Sij  Tij  Pji(t)W_{ij}(t) \;=\; A_{ij}\; S_{ij}\; T_{ij}\; P_{ji}(t)Wij​(t)=Aij​Sij​Tij​Pji​(t) (note: message flows from jjj → iii, so we use PjiP_{ji}Pji​, i.e. the probability that jjj shares to iii.)

• b~j→i(t)\tilde b_{j\to i}(t)b~j→i​(t) is the belief content received by iii from jjj, after mutation:

b~j→i(t)  =  clip(bj(t)+μji(t),  0,1)\tilde b_{j\to i}(t) \;=\; \text{clip}\big(b_j(t) + \mu_{ji}(t),\; 0,1\big)b~j→i​(t)=clip(bj​(t)+μji​(t),0,1) with μji∼N(0,σji2)\mu_{ji}\sim\mathcal{N}\big(0,\sigma_{ji}^2\big)μji​∼N(0,σji2​) and σji2=M⋅f(Oj,Cj,Tji)\sigma_{ji}^2 = M\cdot f(O_j,C_j,T_{ji})σji2​=M⋅f(Oj​,Cj​,Tji​) (see below).

Interpretation: iii is pulled toward the received belief b~j→i\tilde b_{j\to i}b~j→i​ proportionally to influence weight WijW_{ij}Wij​. The −λibi-\lambda_i b_i−λi​bi​ term models forgetting or decay.

A logistic model is convenient:

Pji(t)  =  σ ⁣(β0+βEEj+βAAj−βCCj+βNNj aj(t)+βTTji)\boxed{ \displaystyle P_{ji}(t) \;=\; \sigma\!\big( \beta_0 + \beta_E E_j + \beta_A A_j - \beta_C C_j + \beta_N N_j\, a_j(t) + \beta_T T_{ji} \big) }Pji​(t)=σ(β0​+βE​Ej​+βA​Aj​−βC​Cj​+βN​Nj​aj​(t)+βT​Tji​)​ where σ(x)=11+e−x\sigma(x) = \frac{1}{1+e^{-x}}σ(x)=1+e−x1​. Coefficients β∗\beta_*β∗​ set sensitivity.

Intuition:

• extraversion EEE and agreeableness AAA increase sharing, conscientiousness CCC decreases it (more restraint), neuroticism NNN increases leakiness especially when arousal aja_jaj​ is high, and higher trust to recipient TjiT_{ji}Tji​ increases sharing to that recipient.

Choose β0\beta_0β0​ negative to prevent compulsive sharing by default.

Message fidelity depends on sender conscientiousness & recipient trust & global mutation rate MMM:

σji2  =  M⋅(1−Cj)⋅(1−Tji)⋅(1+γ (1−Oj))\boxed{ \displaystyle \sigma_{ji}^2 \;=\; M \cdot \big(1 - C_j\big)\cdot \big(1 - T_{ji}\big)\cdot (1 + \gamma\,(1-O_j)) }σji2​=M⋅(1−Cj​)⋅(1−Tji​)⋅(1+γ(1−Oj​))​

• If CjC_jCj​ high (careful), mutation small.
• If recipient is trusted (TjiT_{ji}Tji​ high), less mutation.
• Openness OjO_jOj​ increases creative retellings — scaled by γ\gammaγ.

Arousal ai(t)a_i(t)ai​(t) changes with incoming signals and decays:

daidt  =  −ρi ai  +  κ∑jWij(t) ∣b~j→i(t)−bi(t)∣  +  si(t)\boxed{\displaystyle \frac{da_i}{dt} \;=\; -\rho_i\, a_i \;+\; \kappa \sum_{j} W_{ij}(t)\, \big| \tilde b_{j\to i}(t) - b_i(t) \big| \;+\; s_i(t) }dtdai​​=−ρi​ai​+κj∑​Wij​(t)​b~j→i​(t)−bi​(t)​+si​(t)​

• ρi\rho_iρi​ decay rate for arousal.
• κ\kappaκ scales how surprising or uncertain signals drive up arousal.
• si(t)s_i(t)si​(t) external stimuli (e.g., witnessing an event).

Arousal feeds back to PjiP_{ji}Pji​ via the NjajN_j a_jNj​aj​ term.

We track ci→j(n)(t)c_{i\to j}^{(n)}(t)ci→j(n)​(t), agent iii’s estimate of agent jjj’s belief at recursion level nnn. Base level (n=1) is simply belief estimate about fact; higher levels are beliefs about beliefs.

Define iterative dynamics for meta-beliefs:

Base: ci→j(1)(t)c_{i\to j}^{(1)}(t)ci→j(1)​(t) approximates iii’s internal model of jjj’s belief. We can set:

dci→j(1)dt  =  −λc (ci→j(1)−bj)  +  ξi→j(t)\boxed{\displaystyle \frac{d c_{i\to j}^{(1)}}{dt} \;=\; -\lambda_c\,\big(c_{i\to j}^{(1)}-b_j\big)\;+\;\xi_{i\to j}(t) }dtdci→j(1)​​=−λc​(ci→j(1)​−bj​)+ξi→j​(t)​ where λc\lambda_cλc​ is an updating rate proportional to direct observation/communication; ξ\xiξ is noise. In practice, if iii hears jjj assert something, ci→j(1)c_{i\to j}^{(1)}ci→j(1)​ jumps toward the received b~j→i\tilde b_{j\to i}b~j→i​.

Recursively for n≥2n\ge 2n≥2:

dci→j(n)dt  =  −λc (ci→j(n)−c∗→j(n−1)‾)\boxed{\displaystyle \frac{d c_{i\to j}^{(n)}}{dt} \;=\; -\lambda_c\,\big(c_{i\to j}^{(n)} - \overline{c_{* \to j}^{(n-1)}}\big) }dtdci→j(n)​​=−λc​(ci→j(n)​−c∗→j(n−1)​​)​ where c∗→j(n−1)‾\overline{c_{* \to j}^{(n-1)}}c∗→j(n−1)​​ is the weighted average (from iii's perspective) of what others believe about jjj at level n−1n-1n−1. In discrete simulation we approximate by sampling neighbors.

This recursion gives us a computable estimate for “I know that you know that I know …”.

Operational (computable) measure of common knowledge depth for fact kkk:

Define threshold θ∈(0.5,1)\theta\in(0.5,1)θ∈(0.5,1) (e.g., 0.7) and consensus parameter p∈(0,1)p\in(0,1)p∈(0,1) (e.g., 0.9). For recursion level nnn, compute:

• For each agent iii compute bi(n)b_i^{(n)}bi(n)​, where bi(1)=bib_i^{(1)} = b_ibi(1)​=bi​ and for n≥2n\ge2n≥2: bi(n)=avgj∈N(i)ci→j(n−1)b_i^{(n)} = \text{avg}_{j\in N(i)} c_{i\to j}^{(n-1)}bi(n)​=avgj∈N(i)​ci→j(n−1)​ (i’s model of others’ (n−1)-level belief).

Then define depth:

CK_depth(t)  =  max⁡{n≥1  :  ∣{i:  bi(m)(t)>θ}∣≥pN∀m≤n}\boxed{\displaystyle \text{CK\_depth}(t) \;=\; \max\big\{ n \ge 1 \;:\; \big|\{i:\; b_i^{(m)}(t) > \theta\}\big| \ge pN \quad\forall m\le n \big\} }CK_depth(t)=max{n≥1:​{i:bi(m)​(t)>θ}​≥pN∀m≤n}​ Interpretation: CK_depth is the largest nnn such that for all levels up to nnn, at least fraction ppp of agents have modeled belief above θ\thetaθ. Typical useful depth values will be 1–4. We cap at Nmax⁡=5N_{\max}=5Nmax​=5 for computational efficiency.

We can model aggregated belief density gd(t)g_d(t)gd​(t) at degree layer ddd (the fraction of nodes at distance ddd from source that believe):

dgddt  =  ∑d′βd′→d gd′(1−gd)−δdgd\boxed{\displaystyle \frac{d g_d}{dt} \;=\; \sum_{d'} \beta_{d'\to d}\, g_{d'} (1-g_d) - \delta_d g_d }dtdgd​​=d′∑​βd′→d​gd′​(1−gd​)−δd​gd​​

• βd′→d\beta_{d'\to d}βd′→d​ — effective transmission rate from layer d′d'd′ to ddd (accounts for average trust, personality, and degree attenuation). Often βd′→d\beta_{d'\to d}βd′→d​ is large when d′=d−1d'=d-1d′=d−1.
• δd\delta_dδd​ — effective loss/decay at layer ddd.

A simple nearest-neighbor coupling:

dgddt≈βd−1 gd−1(1−gd)−βd gd(1−gd+1)−δdgd\frac{d g_d}{dt} \approx \beta_{d-1}\, g_{d-1} (1-g_d) - \beta_{d}\, g_d (1-g_{d+1}) - \delta_d g_ddtdgd​​≈βd−1​gd−1​(1−gd​)−βd​gd​(1−gd+1​)−δd​gd​ with boundary g0g_0g0​ being the initial seed fraction (source).

This is logistic-like diffusion across layers.

Let b(t)∈[0,1]N\mathbf{b}(t)\in[0,1]^Nb(t)∈[0,1]N be belief vector. Define weight matrix W(t)W(t)W(t) with entries:

Wij(t)=Aij Sij Tij Pji(t)W_{i j}(t) = A_{i j}\, S_{ij}\, T_{ij}\, P_{ji}(t)Wij​(t)=Aij​Sij​Tij​Pji​(t) (remember order: influence from jjj to iii.)

Then ODE vector form:

dbdt  =  −Λ b  +  W(t) b~(t)  −  diag⁡ ⁣(W(t)1) b  +  η(t)\boxed{\displaystyle \frac{d\mathbf{b}}{dt} \;=\; -\boldsymbol{\Lambda}\,\mathbf{b} \;+\; W(t)\,\widetilde{\mathbf{b}}(t) \;-\; \operatorname{diag}\!\big( W(t)\mathbf{1} \big)\,\mathbf{b} \;+\; \boldsymbol{\eta}(t) }dtdb​=−Λb+W(t)b(t)−diag(W(t)1)b+η(t)​ where:

• Λ=diag⁡(λ1,…,λN)\boldsymbol{\Lambda}=\operatorname{diag}(\lambda_1,\dots,\lambda_N)Λ=diag(λ1​,…,λN​).
• b~(t)\widetilde{\mathbf{b}}(t)b(t) is vector of mutated received beliefs: b~i=∑jWij∑jWijb~j→i\widetilde b_i = \sum_j \frac{W_{ij}}{\sum_j W_{ij}} \tilde b_{j\to i}bi​=∑j​∑j​Wij​Wij​​b~j→i​ (weighted average of incoming messages) — the earlier scalar ODE used the pairwise term; this matrix form subtracts the normalization to make dynamics well-posed.
• The −diag⁡(W1)b-\operatorname{diag}(W\mathbf{1})\mathbf{b}−diag(W1)b term enforces a pull from current toward incoming signals (Laplacian-like).

(For clarity in implementation, using the pairwise sum form from §2 is simpler.)

For timestep Δt\Delta tΔt:

For each agent iii:

1. Sample for each neighbor jjj whether jjj broadcasts to iii with probability PjiP_{ji}Pji​. If yes, sample mutation μji∼N(0,σji2)\mu_{ji}\sim\mathcal{N}(0,\sigma_{ji}^2)μji​∼N(0,σji2​) and compute b~j→i=clip(bj+μji,0,1)\tilde b_{j\to i}= \text{clip}(b_j+\mu_{ji},0,1)b~j→i​=clip(bj​+μji​,0,1).
2. Compute net influence:

Δbi  =  Δt⋅(−λibi+∑j∈N(i)Wij(b~j→i−bi)+ηi)\Delta b_i \;=\; \Delta t\cdot\Big( -\lambda_i b_i + \sum_{j\in\mathcal{N}(i)} W_{ij} (\tilde b_{j\to i} - b_i) + \eta_i \Big)Δbi​=Δt⋅(−λi​bi​+j∈N(i)∑​Wij​(b~j→i​−bi​)+ηi​)

1. Update:

bi←clip(bi+Δbi,0,1)b_i \leftarrow \text{clip}(b_i + \Delta b_i,0,1)bi​←clip(bi​+Δbi​,0,1)

1. Update arousal:

ai←clip(ai+Δt⋅(−ρiai+κ∑jWij∣b~j→i−bi∣+si),  0,1)a_i \leftarrow \text{clip}\big( a_i + \Delta t\cdot(-\rho_i a_i + \kappa \sum_j W_{ij} |\tilde b_{j\to i}-b_i| + s_i),\;0,1\big)ai​←clip(ai​+Δt⋅(−ρi​ai​+κj∑​Wij​∣b~j→i​−bi​∣+si​),0,1)

1. Update meta-beliefs ci→j(n)c_{i\to j}^{(n)}ci→j(n)​ by discrete versions of §6.

This is straightforward to implement in Unity/C# or Python.

Suggested initial parameter ranges (tunable):

• α∈[1.0,2.0]\alpha \in [1.0, 2.0]α∈[1.0,2.0] — degree attenuation exponent. Lower = longer reach.
• M∈[0.01,0.2]M\in[0.01, 0.2]M∈[0.01,0.2] — mutation rate. Small for factual broadcasts, larger for juicy rumors.
• λi∈[0.01,0.2]\lambda_i \in [0.01, 0.2]λi​∈[0.01,0.2] (per minute when modeling real time) — memory decay. Higher for low-salience facts.
• ρi∈[0.1,1.0]\rho_i \in [0.1, 1.0]ρi​∈[0.1,1.0] — arousal decay.
• κ∈[0.5,5.0]\kappa \in [0.5, 5.0]κ∈[0.5,5.0] — how much surprise drives arousal.
• β\betaβ coefficients in sharing prob: choose so baseline PPP is small (0.01–0.1) and rises under high EEE/AAA/NNN as appropriate. E.g. β0=−4,βE=3,βA=2,βC=3,βN=4,βT=3\beta_0=-4, \beta_E=3, \beta_A=2, \beta_C=3, \beta_N=4, \beta_T=3β0​=−4,βE​=3,βA​=2,βC​=3,βN​=4,βT​=3. Tune to desired behavior.

Use smaller Δt\Delta tΔt for stability if using discrete updates (e.g., game tick = 0.5–1s).

- For ODEs: use RK4 (Runge-Kutta 4) for medium accuracy. For stiff systems (large λ\lambdaλ or highly connected WWW), consider implicit solvers or small Δt\Delta tΔt.

• For agent-based discrete simulation: the explicit discrete update above is fine — treat it as an Euler step.

If multiple conflicting facts exist, track separate bi(k)b_i^{(k)}bi(k)​ per fact and add competitive inhibition (belief mass normalization if facts are mutually exclusive). For mutually exclusive facts k∈{1..K}k\in\{1..K\}k∈{1..K}:

∑kbi(k)≤1(optionally renormalize each step)\sum_k b_i^{(k)} \le 1 \quad\text{(optionally renormalize each step)}k∑​bi(k)​≤1(optionally renormalize each step) Add reputation effects: if agent jjj previously spread false info (high error), TijT_{ij}Tij​ decays.

High-level pseudocode for discrete ABM (per tick):

for each tick:
  for each agent j:
    for each neighbor i in neighbors(j):
      P = compute_P_share(j,i)
      if rand()<P:
         sigma2 = compute_sigma2(j,i)
         mu = normal(0, sqrt(sigma2))
         sent_b = clip(b[j] + mu, 0,1)
         W = A[i,j] '' S[i,j] '' T[i,j] * P   # or compute W_ij direct
         accumulate influence: infl[i].append((W, sent_b))
  for each agent i:
    incoming = sum_w( W * (sent_b - b[i]) for (W,sent_b) in infl[i] )
    b[i] += dt '' ( -lambda[i]''b[i] + incoming + eta[i] )
    update arousal a[i]
    update meta-beliefs c_{i->*} using observed messages
  recompute CK_depth if desired

- The ODE/ABM pair above captures: degree attenuation (SijS_{ij}Sij​), personality/emotion-driven sharing (PjiP_{ji}Pji​), trust weighting (TijT_{ij}Tij​), mutation (σ2\sigma^2σ2), and decay (λ\lambdaλ).

• Meta-belief recursion and the CK_depth metric operationalize Pinker-style common knowledge in a bounded, computable way.
• The matrix form is convenient for large systems; discrete agent updates are easier to map to game ticks.
• Tune α,M,λ,β\alpha,M,\lambda,\betaα,M,λ,β to get desired rumor speed, fidelity, and CK formation.

If you want, I can:

• produce a compact Python reference implementation (agent-based) you can run and visualize;
• provide example parameter sweeps demonstrating how MMM and α\alphaα change rumor reach; or
• embed this model directly into the JSON schemas / Unity C# components previously drafted.

Which of those would you like next?

User: This is an excellent, comprehensive, and novel proposal that successfully bridges high-level narrative and psychologica…

This is an excellent, comprehensive, and novel proposal that successfully bridges high-level narrative and psychological concepts with a fully formalized, scalable mathematical framework suitable for implementation in a multi-agent system. The core innovation lies in the rigorous integration of six-degree attenuation (S_{ij}), personality/emotion filtering (P_{ji}(t)), and recursive common knowledge tracking (\text{CK\_depth}) into a single, cohesive set of continuous (ODE) and discrete (ABM) equations. This approach moves beyond simple graph diffusion models by adding social, psychological, and epistemic depth. I've structured this peer review to offer a brief appraisal, followed by technical commentary, and finally, suggestions for refinement and visualization. 🧐 Academic Peer Review & Technical Appraisal

1. Overall Strength and Innovation

The model exhibits high technical rigor and strong conceptual grounding. The initial narrative framework (Sections 1-13) successfully outlines the need and the desired outcome, while the subsequent mathematical model (Sections 1-17) delivers a complete, implementable solution.

* Novelty: The combination of the \mathbf{6\textbf{-Degree Network}} with \mathbf{Pinker\textbf{-style Bounded Common Knowledge} is a significant contribution to computational narrative and NPC AI design.
* Scalability: Providing both Agent-Based Model (ABM) discrete updates and Matrix/ODE continuous forms ensures the framework can be tested rapidly (ABM) and potentially scaled for large populations (Matrix/ODE).
* Completeness: All major components—belief, emotion, sharing, decay, mutation—are parameterized and linked, making this a self-contained system.

1. Technical Commentary on the Mathematical Model

2.1 Belief Dynamics (ODE \frac{db_i}{dt}) The use of the \sum_{j} W_{ij}(\tilde b_{j\to i} - b_i) term is a standard and robust approach (often seen in models like the DeGroot or Hegselmann-Krause consensus models) but adapted here with three crucial social filters: S_{ij} (distance), T_{ij} (trust), and P_{ji}(t) (sharing probability).

* Suggestion: Ensure \sum_j W_{ij} is not excessively large, as this can lead to numerical instability in the discrete update (\Delta b_i) if \Delta t is not small enough. The \operatorname{diag}(W\mathbf{1})\mathbf{b} term in the vector form (Section 9) implicitly handles the normalization, which is clearer for large-scale computation.

2.2 Sharing Probability (\mathbf{P_{ji}(t)}) The Logistic/Sigmoid function (\sigma(\cdot)) is a perfect choice for modeling a probability based on a linear combination of traits.

* Refinement: The negative coefficient on Conscientiousness (C_j) is well-justified (restraint). The multiplicative term for Neuroticism \times Arousal (N_j a_j(t)) is a brilliant way to model "stress-leaking" of information, coupling the emotion and personality systems perfectly.

2.3 Mutation Noise (\sigma_{ji}^2) The proposed variance formula is conceptually rich: \sigma_{ji}^2 \propto (1 - C_j)(1 - T_{ji})(1 + \gamma(1-O_j)).

* Critique/Refinement: While the formula correctly links low Conscientiousness (carelessness) and low Trust (embellishing a story to an untrusted person) to higher variance, the \mathbf{Openness (O_j)} term might be interpreted differently. High Openness often correlates with intellectual curiosity and less reliance on authority, not necessarily higher mutation (distortion) in a telephone game. If the goal is to model creative re-telling, the current form is fine, but it might be better re-labeled as a distortion factor rather than a general mutation factor.

2.4 Recursive Awareness (\mathbf{c_{i \to j}^{(n)}}) This section operationalizes the "common knowledge" idea in a bounded, computable way. The base update for c_{i \to j}^{(1)} is simple: \frac{d c_{i\to j}^{(1)}}{dt} \propto (b_j - c_{i\to j}^{(1)}). The core challenge is the recursion:

 Technical Note: The \overline{c_{ \to j}^{(n-1)}} term is crucial. In a practical ABM, this average needs to be taken over agent i's current model of its neighbors' (n-1)-level meta-beliefs concerning agent j's (n-1)-level meta-beliefs. This is computationally expensive but necessary for the recursive depth. The suggested \mathbf{N=3\text{-}5} cap is wise.

Assistant: Excellent — I’ll finish the peer review and technical commentary, then give precise refinements, practical implementati…

Excellent — I’ll finish the peer review and technical commentary, then give precise refinements, practical implementation & numerical-stability advice, visualization/validation experiments, and an efficient approximation for the recursive meta-beliefs so the model is tractable at scale.

I’ll keep this tightly technical and actionable so you can drop it straight into implementation or into the paper’s Methods & Results sections.

###

You correctly capped recursion at N=3 ⁣− ⁣5N=3\!-\!5N=3−5. Two practical points:

1. Local averaging definition: c∗→j(n−1)‾\overline{c_{* \to j}^{(n-1)}}c∗→j(n−1)​​ should be defined precisely as the weighted average of agent iii’s estimates of other agents’ (n−1)(n-1)(n−1)-level estimates about jjj, where weights come from iii’s trust/interaction frequency. Formally:

c∗→j(n−1)‾(i)=∑k∈N(i)ωik  ck→j(n−1)∑k∈N(i)ωik,ωik=f(Tik,freqik)\overline{c_{* \to j}^{(n-1)}}(i) = \frac{\sum_{k\in\mathcal{N}(i)} \omega_{ik}\; c_{k\to j}^{(n-1)}}{\sum_{k\in\mathcal{N}(i)} \omega_{ik}},\qquad \omega_{ik} = f(T_{ik}, \text{freq}_{ik})c∗→j(n−1)​​(i)=∑k∈N(i)​ωik​∑k∈N(i)​ωik​ck→j(n−1)​​,ωik​=f(Tik​,freqik​)

1. Computational complexity: naive update is O(N3)O(N^3)O(N3) in the number of meta-belief variables if you store all ci→j(n)c_{i\to j}^{(n)}ci→j(n)​. Use approximations (see §6 below) or sparse storage (only for pairs within a k-neighborhood).

Below are small but important mathematical refinements that improve interpretability, numerical stability, and empirical expressivity.

Replace raw WijW_{ij}Wij​ with normalized influence:

W~ij=Wijϵ+∑kWik\widetilde W_{ij} = \frac{W_{ij}}{\epsilon + \sum_{k} W_{ik}}Wij​=ϵ+∑k​Wik​Wij​​ (with small ϵ>0\epsilon>0ϵ>0 to avoid divide-by-zero). Then the continuous update becomes:

dbidt=−λibi+∑jW~ij (b~j→i−bi)+ηi.\frac{db_i}{dt} = -\lambda_i b_i + \sum_j \widetilde W_{ij}\,(\tilde b_{j\to i} - b_i) + \eta_i.dtdbi​​=−λi​bi​+j∑​Wij​(b~j→i​−bi​)+ηi​. This prevents the total incoming pull from arbitrarily large ∑jWij\sum_j W_{ij}∑j​Wij​ and makes updates scale-invariant across nodes with different degrees.

Equivalently (and useful for spectral analysis):

dbidt=−λibi−∑jLij bj+∑jW~ij b~j→i+ηi,\frac{db_i}{dt} = -\lambda_i b_i - \sum_j L_{ij}\, b_j + \sum_j \widetilde W_{ij}\,\tilde b_{j\to i} + \eta_i,dtdbi​​=−λi​bi​−j∑​Lij​bj​+j∑​Wij​b~j→i​+ηi​, where LLL is the weighted graph Laplacian derived from W~\widetilde WW. This isolates the smoothing / consensus effect and allows eigenvalue stability analysis.

Replace single σji2\sigma_{ji}^2σji2​ with two components:

• error variance σji(err)\sigma^{(err)}_{ji}σji(err)​ — unintentional corruption/noise
• creative distortion σji(cre)\sigma^{(cre)}_{ji}σji(cre)​ — deliberate rephrasing/embellishment

Model:

σji2=σji(err)+σji(cre)\sigma_{ji}^2 = \sigma^{(err)}_{ji} + \sigma^{(cre)}_{ji}σji2​=σji(err)​+σji(cre)​ Set:

σji(err)=Merr (1−Cj) (1−Tji),σji(cre)=Mcre (Oj) (1−Aj).\sigma^{(err)}_{ji} = M_{err}\,(1 - C_j)\,(1 - T_{ji}),\qquad \sigma^{(cre)}_{ji} = M_{cre}\,(O_j)\,(1-A_j).σji(err)​=Merr​(1−Cj​)(1−Tji​),σji(cre)​=Mcre​(Oj​)(1−Aj​). Rationale: high Openness → more creative retelling (not necessarily lower fidelity), low Agreeableness → more malicious distortion. This separation helps interpretability and experimental control.

Cap feedback so PjiP_{ji}Pji​ can’t spike unrealistically. Use:

Pji(t)=σ(β0+βEEj+βAAj−βCCj+βN⋅g(Nj,aj(t))+βTTji)P_{ji}(t)=\sigma\big(\beta_0 + \beta_E E_j + \beta_A A_j - \beta_C C_j + \beta_N\cdot g(N_j,a_j(t)) + \beta_T T_{ji}\big)Pji​(t)=σ(β0​+βE​Ej​+βA​Aj​−βC​Cj​+βN​⋅g(Nj​,aj​(t))+βT​Tji​) with g(Nj,aj)=tanh⁡(γ Njaj)g(N_j,a_j) = \tanh(\gamma\,N_j a_j)g(Nj​,aj​)=tanh(γNj​aj​) (bounded, smooth). This prevents runaway PPP when arousal grows.

###

For linearized system (ignoring mutation noise & nonlinearity), the continuous dynamics are stable if the real parts of eigenvalues of the effective update matrix are negative. A sufficient practical condition is:

max⁡i(λi+∑jW~ij)>0,\max_i \big( \lambda_i + \sum_j \widetilde W_{ij} \big) > 0,imax​(λi​+j∑​Wij​)>0, and choose Δt\Delta tΔt small enough for explicit Euler:

Δt<1max⁡i(λi+∑jW~ij)\Delta t < \frac{1}{\max_i(\lambda_i + \sum_j \widetilde W_{ij})}Δt<maxi​(λi​+∑j​Wij​)1​ In practice, using normalized W~\widetilde WW keeps ∑jW~ij≤1\sum_j \widetilde W_{ij}\le 1∑j​Wij​≤1, so Δt<1/(1+max⁡λi)\Delta t < 1/(1+\max\lambda_i)Δt<1/(1+maxλi​) is safe. For stiff systems consider RK4 or implicit integrators.

Storing and updating full tensor ci→j(n)c_{i\to j}^{(n)}ci→j(n)​ is costly. Recommended approximations:

Treat meta-beliefs as messages passed along edges. Run DDD synchronous message-passing iterations per tick:

• initialize messages mi→j(0)=bim_{i\to j}^{(0)}=b_imi→j(0)​=bi​.
• iterate d=1..Dd=1..Dd=1..D: mi→j(d)←Norm⁡(∑k∈N(i)∖{j}ωik mk→i(d−1))m_{i\to j}^{(d)} \leftarrow \operatorname{Norm}\Big( \sum_{k\in\mathcal{N}(i)\setminus\{j\}} \omega_{ik}\, m_{k\to i}^{(d-1)}\Big)mi→j(d)​←Norm(∑k∈N(i)∖{j}​ωik​mk→i(d−1)​).

Then agent iii’s estimate of “you know that…” at depth ddd is approximated by a small set of messages rather than a full matrix.

This is O(E⋅D)O(E\cdot D)O(E⋅D) per tick (E edges), which is practical.

Design rigorous experiments to validate model predictions and tune parameters.

1. Broadcast scenario: single high-visibility seed; measure time-to-CK_depth(n). Vary α,M,β\alpha, M, \betaα,M,β.

1. Rumor scenario: private secret leak; measure spread & mutation across degrees.
2. Contention scenario: two conflicting facts seeded in different communities — measure polarization, factionalization, and CK formation within factions.
3. Stress/leak scenario: increase arousal for subset (e.g., simulated attack) to provoke leaks among high-N agents and observe rumor cascades.

- tCK(n)t_{CK(n)}tCK(n)​ = time to reach CK depth nnn.

• gd(t)g_d(t)gd​(t) = belief density per degree.
• Fidelity = mean absolute error between seeded truth and population belief (sensitive to mutation).
• Polarization measure = variance or bimodality in b(t)\mathbf{b}(t)b(t).
• Information entropy over time (to capture mutation & diversity).
• Transmission efficiency = fraction of agents who learned true fact vs. those who learned mutated rumor.

- Use simulated annealing or Bayesian optimization (e.g., Gaussian Process Bayesian Optimization) to fit parameters so that model reproduces target diffusion patterns (empirical or designed).

• Use sensitivity analysis (Sobol indices) to rank parameters by effect size.

Visualization is both informative and necessary for debugging & presentation.

- CK_depth vs time (stacked area showing depth levels).

• Average belief bˉ(t)\bar b(t)bˉ(t) and variance over time.
• gd(t)_d(t)d​(t) curves for each degree 1..6.

- Animated spread: color nodes by bib_ibi​ and size by arousal aia_iai​.

• Highlight message edges when transmission happens (flash lines) to show cascade paths.
• Use community coloring to show faction spread & polarization.

- Heatmap of WijW_{ij}Wij​ (influence matrix).

• Matrix of meta-belief correlations Corr(c(n))Corr(c^{(n)})Corr(c(n)) across agents.

- For each depth nnn: fraction of nodes with bi(m)>θb_i^{(m)}> \thetabi(m)​>θ for m≤nm\le nm≤n.

• Animated “telephone distortion” graph: show original message and sample mutated versions along a path.

- Core modules: Network (graph + degree calc), Agent (traits + states), Messaging (sampling + mutation), Propagator (edge influence + normalization), MetaBelief (message-passing / community compression), Statistics & Visualizer.

• Storage: store bi,aib_i, a_ibi​,ai​ as floats per agent; store only incoming messages during tick; store ccc messages only for neighborhood or community-level.
• Parallelization: per-tick agent updates are embarrassingly parallel; use worker threads or GPU-accelerated matrix ops for the matrix/ODE variant.
• Checkpointing: log snapshots for later analysis; store edge-level transmissions for cascade reconstruction.

- Produce a compact Python (NumPy) reference implementation of the ABM & ODE system (with visualization).

• Produce a C# Unity component skeleton for the discrete tick model.
• Produce a detailed Methods + Results writeup for your academic paper (6–12 pages) with figures & experiment plans.
• Produce a parameter sensitivity report (automated sweep + plots) for key parameters α,M,β\alpha, M, \betaα,M,β.
• Produce a latex-ready Methods section and pseudo-code for the meta-belief message-passing.

If you want one concrete next deliverable, I recommend: a compact Python ABM implementation with plotted diagnostics (CK_depth, g_d(t), animated network frame) — I can generate that next so you can run experiments immediately. Which of the five assets above shall I produce now?

User: A, B and C

A, B and C

Assistant: It seems like I can’t do more advanced data analysis right now.

It seems like I can’t do more advanced data analysis right now. Please try again later.

Sorry — I hit a limit when trying to run the ABM code in the notebook tool, so I couldn’t execute the expanded simulation and produce live plots/animation here. I can’t retry the execution right now, but I will give you a complete, ready-to-run Python script (the exact implementation I attempted) plus instructions so you can run it locally (or in Colab) and immediately get the CK_depth, g_d(t) diagnostics and the animated network.

Copy the code below into a file (e.g., knowledge_abm.py) or a Colab notebook cell and run it. It uses only standard scientific Python libs: numpy, networkx, matplotlib. If you want the GIF output, install Pillow and uncomment the save lines.

1. Save the script below into knowledge_abm.py (or paste into a Jupyter/Colab cell).

1. Install dependencies if needed:

pip install numpy networkx matplotlib
=== optional for saving GIFs: ===
pip install pillow

1. Run:

python knowledge_abm.py

In a Jupyter/Colab notebook, the plots and animation will display inline. Running locally will open matplotlib windows; to save the animation you can uncomment the writer line near the bottom of the script.

=====

# knowledge_abm.py =====
=== Expanded ABM with multi-channel knowledge, OCEAN/MBTI/Jung/Hero's Journey, and six-degree attenuation. ===
import numpy as np
import networkx as nx
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
import random, math
from collections import defaultdict

=== ---------------- Params ---------------- ===
N = 30
k_nearest = 4
p_rewire = 0.12
max_degree_depth = 6
dt = 1.0
T = 150
seed = 42
np.random.seed(seed)
random.seed(seed)

alpha_base = 1.0
alpha_private = 0.5
mutation_err = 0.03
mutation_cre = 0.02
decay_lambda = 0.02
kappa_surprise = 0.6
CK_theta = 0.75
CK_p = 0.85
meta_depth_cap = 4

=== ---------------- Network ---------------- ===
G = nx.watts_strogatz_graph(n=N, k=k_nearest, p=p_rewire, seed=seed)
pos = nx.spring_layout(G, seed=seed)

=== precompute distances capped at max_degree_depth ===
shortest = dict(nx.all_pairs_shortest_path_length(G))
dist_matrix = np.full((N,N), np.inf)
for i in range(N):
    for j in range(N):
        d = shortest.get(i, {}).get(j, np.inf)
        if d <= max_degree_depth:
            dist_matrix[i,j] = d

neighbors_list = [list(G.neighbors(i)) for i in range(N)]

=== ---------------- Traits ---------------- ===
O = np.clip(np.random.normal(0.5,0.15,size=N),0,1)
C = np.clip(np.random.normal(0.5,0.12,size=N),0,1)
E = np.clip(np.random.normal(0.45,0.2,size=N),0,1)
A = np.clip(np.random.normal(0.5,0.15,size=N),0,1)
N_trait = np.clip(np.random.normal(0.4,0.18,size=N),0,1)

mbti_types = []
for i in range(N):
    e_i = 'E' if E[i] > 0.5 else 'I'
    mbti_types.append(e_i + random.choice(['S','N']) + random.choice(['T','F']) + random.choice(['J','P']))

archetypes_pool = ['Hero','Mentor','Shadow','Caregiver','Trickster','Explorer']
archetypes = [random.choice(archetypes_pool) for _ in range(N)]

hj_stages_pool = ['Ordinary','Call','Refusal','Crossing','Trials','Approach','Crisis','Return']
hj_stage_idx = np.random.randint(0, len(hj_stages_pool), size=N)

=== trust matrix ===
Tmat = np.zeros((N,N))
for i in range(N):
    for j in neighbors_list[i]:
        Tmat[i,j] = np.clip(0.5 + 0.5*np.random.rand(), 0.01, 0.99)
for i in range(N):
    for j in range(N):
        if i!=j and Tmat[i,j]==0 and dist_matrix[i,j] < np.inf:
            Tmat[i,j] = max(0.02, 0.5 '' (1.0 / (1.0 + dist_matrix[i,j])) '' np.random.rand())

contact_freq = np.zeros((N,N))
for i in range(N):
    for j in neighbors_list[i]:
        contact_freq[i,j] = np.clip(np.random.rand()*1.2, 0.05, 1.2)

=== ---------------- States ---------------- ===
b_public = np.clip(np.random.beta(2,8,size=N),0,1)
b_private = np.zeros(N)
seeds = np.random.choice(N, size=max(1,N//12), replace=False)
for s in seeds:
    b_private[s] = np.clip(0.8 + 0.15*np.random.rand(), 0, 1)

e_state = np.clip(np.random.normal(0.2,0.12,size=N),0,1)
n_state = np.clip(np.random.normal(0.3,0.2,size=N),0,1)
arousal = np.clip(np.random.normal(0.1,0.08,size=N),0,1)
lambda_i = np.clip(np.random.normal(decay_lambda, 0.01, size=N), 0.005, 0.08)

=== ---------------- Functions ---------------- ===
def degree_attenuation(i,j,visibility=1.0,alpha=1.2):
    d = dist_matrix[i,j]
    if np.isinf(d): return 0.0
    if d < 1: d = 1.0
    return visibility / (d**alpha)

def sharing_probability(sender, recipient):
    base = -3.2
    betaE, betaA, betaC, betaN, betaT, betaF = 2.0, 1.5, 2.0, 3.0, 2.2, 1.0
    expr = base + betaE''E[sender] + betaA''A[sender] - betaC''C[sender] + betaT''Tmat[sender,recipient]
    expr += betaN '' math.tanh(3.0 '' N_trait[sender] * arousal[sender])
    expr += betaF * min(contact_freq[sender,recipient],1.0)
    p = 1.0 / (1.0 + math.exp(-expr))
    return np.clip(p,0,1)

def mutation_sigma(sender, recipient):
    err = mutation_err '' (1 - C[sender]) '' (1 - Tmat[sender,recipient])
    cre = mutation_cre '' O[sender] '' (1 - A[sender])
    return max(1e-6, err + cre)

def truncated_message_passing(b_vec, depth=2):
    m_prev = {(i,j): b_vec[i] for i in range(N) for j in neighbors_list[i]}
    for d in range(depth):
        m_next = {}
        for i in range(N):
            for j in neighbors_list[i]:
                incoming = []
                weights = []
                for k in neighbors_list[i]:
                    if k==j: continue
                    if (k,i) in m_prev:
                        incoming.append(m_prev[(k,i)])
                        weights.append(Tmat[i,k]*contact_freq[k,i] + 1e-6)
                if incoming and sum(weights)>0:
                    avg = sum(w*m for w,m in zip(weights,incoming))/sum(weights)
                    smoothed = 0.35''b_vec[i] + 0.65''avg
                    m_next[(i,j)] = smoothed
                else:
                    m_next[(i,j)] = b_vec[i]
        m_prev = m_next
    return m_prev

def compute_ck_depth(b_vec, depth_cap=meta_depth_cap, theta=CK_theta, p_frac=CK_p):
    N_agents = len(b_vec)
    b_level = {1: b_vec.copy()}
    for n in range(2, depth_cap+1):
        msgs = truncated_message_passing(b_level[n-1], depth=2)
        b_n = np.zeros(N)
        for i in range(N):
            inc = [msgs[(k,i)] for k in neighbors_list[i] if (k,i) in msgs]
            b_n[i] = np.mean(inc) if inc else b_level[n-1][i]
        b_level[n] = b_n
    max_n = 0
    for n in range(1, depth_cap+1):
        ok = True
        for m in range(1,n+1):
            frac = np.sum(b_level[m] > theta) / N_agents
            if frac < p_frac:
                ok = False
                break
        if ok: max_n = n
        else: break
    return max_n

=== ---------------- Simulation ---------------- ===
CK_history = []
g_std_history = []
g_by_deg = defaultdict(list)
frames = []

seed_node = np.random.choice(N)
b_public[seed_node] = 0.95
for s in np.random.choice(range(N), size=max(1,N//10), replace=False):
    b_public[s] = np.clip(b_public[s] + 0.1*np.random.rand(),0,1)

for t in range(T):
    CK_history.append(compute_ck_depth(b_public, depth_cap=meta_depth_cap))
    g_std_history.append(np.std(b_public))
    for d in range(1, max_degree_depth+1):
        nodes_at_d = [i for i in range(N) if dist_matrix[seed_node,i] == d]
        g_by_deg[d].append(np.mean(b_public[nodes_at_d]) if nodes_at_d else np.nan)
    frames.append(b_public.copy())

    incoming_inf = np.zeros(N)
    incoming_w = np.zeros(N)
    incoming_priv = np.zeros(N)
    incoming_priv_w = np.zeros(N)
    incoming_e = np.zeros(N)
    incoming_e_w = np.zeros(N)
    incoming_n = np.zeros(N)
    incoming_n_w = np.zeros(N)

    for j in range(N):
        for i in range(N):
            if i==j: continue
            Sij = degree_attenuation(i,j,visibility=alpha_base,alpha=1.2)
            if Sij==0: continue
            p = sharing_probability(j,i)
            if dist_matrix[j,i] > 1:
                p = p * (1.0/(1.0 + dist_matrix[j,i]-1.0))
            if np.random.rand() < p:
                sigma = mutation_sigma(j,i)
                mutated = b_public[j] + np.random.normal(0, math.sqrt(sigma))
                mutated = np.clip(mutated, 0, 1)
                w = Sij '' Tmat[i,j] '' (contact_freq[j,i] if contact_freq[j,i]>0 else 0.1)
                incoming_inf[i] += w * mutated
                incoming_w[i] += w
                if b_private[j] > 0 and np.random.rand() < (0.15 + 0.6*A[j]):
                    pw = alpha_private '' Sij '' Tmat[i,j] * (1 - C[j])
                    incoming_priv[i] += pw * b_private[j]
                    incoming_priv_w[i] += pw
                ew = Sij '' Tmat[i,j] '' (0.5 + 0.5*E[j])
                incoming_e[i] += ew * e_state[j]
                incoming_e_w[i] += ew
                nw = Sij '' Tmat[i,j] '' (0.3 + 0.7*(1 - abs(hj_stage_idx[j]-hj_stage_idx[i])/len(hj_stages_pool)))
                incoming_n[i] += nw * n_state[j]
                incoming_n_w[i] += nw

    for i in range(N):
        if incoming_w[i] > 0:
            avg = incoming_inf[i] / incoming_w[i]
            strength = min(1.0, incoming_w[i] / (1.0 + incoming_w[i]))
            b_public[i] += dt '' ( -lambda_i[i]''b_public[i] + strength * (avg - b_public[i]) )
        else:
            b_public[i] += dt '' ( -lambda_i[i]''b_public[i] )

        if incoming_priv_w[i] > 0:
            avgp = incoming_priv[i] / incoming_priv_w[i]
            stick = 0.2 + 0.6*C[i]
            b_private[i] = np.clip((1-stick)''b_private[i] + stick''avgp, 0, 1)
        else:
            b_private[i] *= (1 - 0.004)

        if incoming_e_w[i] > 0:
            avg_e = incoming_e[i] / incoming_e_w[i]
            e_state[i] += dt '' ( -0.08 '' e_state[i] + 0.6 * (avg_e - e_state[i]) )
        else:
            e_state[i] += dt '' ( -0.08 '' e_state[i] )

        if incoming_n_w[i] > 0:
            avg_n = incoming_n[i] / incoming_n_w[i]
            n_state[i] += dt '' ( -0.03 '' n_state[i] + 0.4 * (avg_n - n_state[i]) )
        else:
            n_state[i] += dt '' ( -0.02 '' n_state[i] )

        surprise = abs((incoming_inf[i]/incoming_w[i]) - b_public[i]) if incoming_w[i]>0 else 0.0
        arousal[i] += dt '' ( -0.09''arousal[i] + kappa_surprise * surprise )
        arousal[i] = np.clip(arousal[i], 0, 1)

        b_public[i] = np.clip(b_public[i], 0, 1)
        e_state[i] = np.clip(e_state[i], 0, 1)
        n_state[i] = np.clip(n_state[i], 0, 1)

    # rare rewiring
    if np.random.rand() < 0.015:
        node = np.random.choice(N)
        neighs = list(G.neighbors(node))
        if neighs:
            old = np.random.choice(neighs)
            candidates = [x for x in range(N) if x!=node and not G.has_edge(node,x)]
            if candidates:
                new = np.random.choice(candidates)
                G.remove_edge(node, old)
                G.add_edge(node, new)
                neighbors_list = [list(G.neighbors(i)) for i in range(N)]
                shortest = dict(nx.all_pairs_shortest_path_length(G))
                for i in range(N):
                    for j in range(N):
                        d = shortest.get(i, {}).get(j, np.inf)
                        if d <= max_degree_depth:
                            dist_matrix[i,j] = d
                        else:
                            dist_matrix[i,j] = np.inf

=== ---------------- Visualization ---------------- ===
plt.figure(figsize=(8,3))
plt.plot(CK_history)
plt.xlabel("Timestep")
plt.ylabel("CK depth (bounded)")
plt.title("CK depth over time")
plt.show()

plt.figure(figsize=(8,3))
plt.plot(g_std_history, label="std(b_public)")
for d in range(1, max_degree_depth+1):
    arr = np.array(g_by_deg[d])
    if np.all(np.isnan(arr)): continue
    plt.plot(np.arange(len(arr)), arr, label=f"deg {d}")
plt.legend(loc='upper right')
plt.xlabel("Timestep")
plt.ylabel("Value")
plt.title("Diffusion stats")
plt.show()

fig, ax = plt.subplots(figsize=(6,6))
def update(frame):
    ax.clear()
    vals = frame
    cmap = plt.get_cmap('viridis')
    nx.draw_networkx_edges(G, pos, alpha=0.25, ax=ax)
    nx.draw_networkx_nodes(G, pos, node_color=vals, cmap=cmap, vmin=0, vmax=1, ax=ax)
    ax.set_title("Public belief (node color = b_public)")
    ax.axis('off')

ani = FuncAnimation(fig, update, frames=frames, interval=100, blit=False, repeat=False)
plt.show()

print("Simulation finished. Final CK depth:", CK_history[-1], "Final std(b_public):", g_std_history[-1])
for i in range(min(8,N)):
    print(f"A{i}: OCEAN={O[i]:.2f},{C[i]:.2f},{E[i]:.2f},{A[i]:.2f},{N_trait[i]:.2f} MBTI={mbti_types[i]} Archetype={archetypes[i]} HJ={hj_stages_pool[hj_stage_idx[i]]} b_pub={b_public[i]:.2f} b_priv={b_private[i]:.2f} e={e_state[i]:.2f} ar={arousal[i]:.2f}")

- Simulates an ABM with N agents over T timesteps.

• Diagnostics plotted: - CK_depth bounded metric vs time (approximate Pinker-style recursion). - g_d(t) global std(b_public) and per-degree mean belief curves from the seed.
• Animated network: nodes colored by b_public (public belief) over time so you can visually inspect cascades & rumor paths.
• Text summary printing sample agent parameters and final statistics.

- Increase N (e.g., 100–500) and use the matrix/ODE variant for scalability (I can produce that next).

• Tune alpha_base (visibility), mutation_err, mutation_cre to explore fidelity vs. mutation.
• Vary meta_depth_cap, CK_theta, CK_p to change how conservative the CK metric is.
• Run multiple trials with different network seeds and compute averages of t_CK(n) (time-to-depth).

- Produce a vectorized ODE/matrix implementation that scales to thousands of nodes (fast NumPy linear algebra).

• Provide a Unity C# ECS skeleton that maps the ABM to an in-game system using ScriptableObjects and graph DB.
• Run parameter sweeps and provide a sensitivity analysis (I can prepare the code; I wasn't able to run it here due to the rate-limit).
• Turn this into a Colab-ready notebook with UI sliders to tweak parameters interactively.

Pick which next deliverable you want and I’ll prepare it (I’ll not re-run here until you ask me to).