Editing Openai/693b7cce-38f4-800d-92f7-5e56a467bfa7 (section)

== 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=⎩⎨⎧dijαV0if 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=−λibi+j=1∑NWij(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)=AijSijTijPji(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−λibi 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+βEEj+βAAj−βCCj+βNNjaj(t)+βTTji)
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=−ρiai+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_jNjaj 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′→dgd′(1−gd)−δdgd
* 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−1gd−1(1−gd)−βdgd(1−gd+1)−δdgd
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)=AijSijTijPji(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∑jWijWijb~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:
# 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).
# 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⋅(−λibi+j∈N(i)∑Wij(b~j→i−bi)+ηi)
# 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)
# 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⋅(−ρiai+κj∑Wij∣b~j→i−bi∣+si),0,1)
# 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): ===

<syntaxhighlight>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

</syntaxhighlight>

=== To know how quickly a fact becomes common knowledge, simulate and track CK_depth(t) (§7). Watch for CK_depth reaching 3 or more — indicates strong recursive public awareness. You can also define time-to-common-knowledge tCK(n)t_{CK(n)}tCK(n) as first time CK_depth ≥n\ge n≥n. ===

=== One seeded broadcast node (official) sets ηseed(t)=1\eta_{seed}(t)=1ηseed(t)=1 for a short duration. With V=1V=1V=1, α=1.3\alpha=1.3α=1.3, β\betaβ as above, M=0.02M=0.02M=0.02, average trust 0.6, you should see fast ramp-up in degrees 1–2 within few ticks, slower infiltration to 3–6, and CK_depth hitting 2–3 if reinforcement (re-broadcasts) occurs. ===

=== - 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
# 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.
# 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:
# 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)ωikck→j(n−1),ωik=f(Tik,freqik)
# 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+∑kWikWij
(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=−λibi+j∑Wij(b~j→i−bi)+ηi.
This prevents the total incoming pull from arbitrarily large ∑jWij\sum_j W_{ij}∑jWij 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=−λibi−j∑Lijbj+j∑Wijb~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+βEEj+βAAj−βCCj+βN⋅g(Nj,aj(t))+βTTji)
with g(Nj,aj)=tanh⁡(γ Njaj)g(N_j,a_j) = \tanh(\gamma\,N_j a_j)g(Nj,aj)=tanh(γNjaj) (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+∑jWij)1
In practice, using normalized W~\widetilde WW keeps ∑jW~ij≤1\sum_j \widetilde W_{ij}\le 1∑jWij≤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.

===== For nodes with extremely high degree, compute influence via sparse sampling of neighbors each tick (subsample k neighbors proportional to ωik\omega_{ik}ωik), or precompute aggregated group influences (community-level summarization). =====

==== 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}ωikmk→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.

===== Compress agents into communities (via modularity / clustering). Track meta-beliefs at community level, and approximate individual-level meta-beliefs from community aggregates. This reduces complexity from O(N2)O(N^2)O(N2) to O(C2)O(C^2)O(C2) where C≪NC\ll NC≪N. =====

===== Each agent maintains meta-belief estimates only for a small random sample of other agents; aggregate statistics approximate global CK_depth. This is efficient if CK metric only needs population-level thresholds. =====

==== 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. =====
# Rumor scenario: private secret leak; measure spread & mutation across degrees.
# Contention scenario: two conflicting facts seeded in different communities — measure polarization, factionalization, and CK formation within factions.
# 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). =====
# Install dependencies if needed:

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

</syntaxhighlight>
# Run:

<syntaxhighlight>python knowledge_abm.py

</syntaxhighlight>

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.

===== <syntaxhighlight lang="python"># 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}")

</syntaxhighlight>

===== - 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).