Mean-field weight-noise invariance conjecture

Let the assumptions of Theorem~ hold, with the dynamics of $(\bm{X}_i(t))_{i\in[N]}$ and $(\bm{w}_{i,j}(t))_{i,j\in[N]}$ governed by the SDEs with weight-noise intensity $\eta>0$. Let $(\overline{X}(t),\overline{w}(t))$ solve the original McKean–Vlasov SDEs, without the $\eta$ term, with initial data from the same independent-mixture sampling. Mean-field weight-noise invariance conjecture. The estimates of the article still hold; in particular, for some constant $C(t)>0$ depending on $t$ and the bounds in Assumptions~ and,

$$\mathbb{E}\bigg[\delta_{W_1,\square}\bigg(\Big((\bm{X}_{i}(t))_{i\in[N]},(\bm{w}_{i,j}(t))_{i,j\in[N]},[N]\Big),\Big(\overline{X}(t),\overline{w}(t),(\Omega\times\widehat{\Omega})\Big)\bigg)\bigg]\leq\frac{C(t)}{\sqrt{\log N}}.$$

The conjecture asserts that independent edge noise has no effect on the limiting McKean–Vlasov dynamics under mean-field scaling, apart from the stated finite-size error.