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

=== 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.