发表机构
Northwestern University（西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出在线通信复形信息流的有限维上链模型，证明节点意见收敛至共识、边流收敛至初始调和投影，还探讨线性阻尼与非线性饱和的修正效果。
AI 中文摘要
我们针对在线通信复形上的信息流提出了一种有限维上链模型。节点变量代表议题立场，边变量代表经独立建模的带符号信息流。耦合关系通过上边缘算子 $d_0$ 与1-上链霍奇拉普拉斯算子 $\triangle_1=d_0d_0^*+d_1^*d_1$ 表示。我们证明了平均意见守恒、李雅普诺夫能量定律，以及收敛至由边流初始调和投影确定的平衡态。具体而言，对于任意初始条件，节点意见会收敛至共识，而边流则收敛至 $P_{\boldsymbol{\textit{H}}^1}u_0$。因此，平凡一阶上同调意味着整个边流变量衰减，而非平凡一阶上同调仅在初始边流具有非零调和投影时，才为非零残余环流提供容量。该模型由此确立：节点共识并不必然意味着经独立表示的边流变量衰减；它并未对行为回音室的形成、强化或放大进行建模。我们还考虑了线性阻尼与有界非线性饱和作为修正项，这些修正项在所述假设下可消除持续边流。
英文摘要
We propose a finite-dimensional cochain model for information flow on an online communication complex. Node variables represent issue positions, while edge variables represent independently modelled signed information flow. The coupling is written in terms of the coboundary operator $d_0$ and the $1$-cochain Hodge Laplacian $Δ_1=d_0d_0^*+d_1^*d_1$. We prove conservation of the mean opinion, a Lyapunov energy law, and convergence to an equilibrium determined by the initial harmonic projection of the edge flow. In particular, node opinions converge to consensus for every initial condition, whereas the edge flow converges to $P_{\mathcal H^1}u_0$. Thus, trivial first cohomology implies decay of the entire edge-flow variable. Nontrivial first cohomology supplies a topological capacity for residual edge circulation. A particular trajectory has a nonzero residual circulation if and only if $P_{\mathcal H^1}u_0\neq 0$, in which case the limiting edge flow is $P_{\mathcal H^1}u_0$. The model therefore establishes that node consensus need not imply decay of an independently represented edge-flow variable; it does not model the formation, reinforcement, or amplification of behavioral echo chambers. We also consider linear damping and a bounded nonlinear saturation as modifications that remove persistent edge flow under the stated assumptions.
