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

==== 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.