Consensus and Persistent Harmonic Edge Circulation in a Hodge-Theoretic Model of Networked Information Flow

Abstract

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 $\Delta_1=d_0d_0^*+d_1^*d_1$. 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.