Selective Adaptation of Beliefs and Communication on Cellular Sheaves

TL;DR

This paper extends the discourse-sheaf framework for opinion dynamics by introducing directional stubbornness and selective learning, enabling partial rigidity in both beliefs and expressions. It develops two parallel forced-sheaf formulations: stubborn opinions lead to a sheaf Poisson problem on the sheaf of free opinions, while stubborn expressions induce diffusion on an auxiliary sheaf of free structures, with equilibria characterized as affine-projection solutions. When beliefs and expressions evolve jointly, four constraint scenarios are analyzed, with convergence guaranteed under suitable conditions and conservation laws preventing degenerate consensus. Timescale separation yields stagnation bounds that quantify when rapid rhetorical adaptation masks stable beliefs or when flexible beliefs align with rigid communication norms, offering a principled view of surface harmony versus genuine accommodation.

Abstract

We extend opinion dynamics on discourse sheaves to incorporate "directional stubbornness": agents may hold fixed positions in specified directions of their opinion stalk while remaining flexible in others. This converts the equilibrium problem from harmonic extension to a forced sheaf equation: the free-opinion component satisfies a sheaf Poisson equation with forcing induced by the clamped directions. We develop a parallel theory for "selective learning" of expression policies. When only a designated subset of incidence maps may adapt, the resulting gradient flow is sheaf diffusion on an auxiliary structure sheaf whose global sections correspond to sheaf structures making a fixed opinion profile publicly consistent. For joint evolution of beliefs and expressions, we give conditions (and regularized variants) guaranteeing convergence to nondegenerate equilibria, excluding spurious agreement via vanishing opinions or trivialized communication maps. Finally, we derive stagnation bounds in terms of the rate ratio between opinion updating and structural adaptation, quantifying when rapid rhetorical accommodation masks nearly unchanged beliefs, and conversely when flexible beliefs conform to rigid communication norms.

Figures (7)

• Figure 1: Sheaf diffusion on a discourse sheaf. Left: Initial opinion distribution $x(0)$ with discrepancy $\|\delta x(0)\|^2 = 6$. Right: The limit $x^\infty = P_{H^0}(x(0))$, the orthogonal projection onto the space of global sections. At equilibrium, all expressed opinions agree: both endpoints of each edge map to the same value in the discourse space ($-2$ on the top edge, $-1$ on the right, $1$ on the bottom, and $(1,0)^T$ on the left).
• Figure 2: Constrained diffusion with a partially stubborn agent. Agent $v_4$ holds a fixed opinion in the first coordinate (red), while all other opinions evolve freely. Left: Initial state with $\|\delta x(0)\|^2 = 5$. Right: Equilibrium with $\|\delta x^\infty\|^2 = 1$. The stubborn direction at $v_4$ remains unchanged at $1$. Edges $e_{12}$, $e_{23}$, and $e_{34}$ achieve perfect agreement, but edge $e_{41}$ retains residual discrepancy $(0, -1)^T$ because $v_4$'s stubbornness prevents full consensus.
• Figure 3: Partial learning with stubborn expressions. Opinions $x$ are fixed throughout; restriction maps from $v_4$ (red) are frozen while the remaining maps (black) adapt. Left: Initial configuration with $\|\delta x\|^2 = 5$. Right: Equilibrium with $\|\delta x\|^2 = 1$. Agents $v_1$, $v_2$, $v_3$ adjust their expression policies to accommodate $v_4$'s rigid communication. Edge $e_{34}$ retains residual discrepancy $1$ because $x_{v_3} = 0$ makes $v_3$'s expressed opinion vanish regardless of its restriction map, while $v_4$'s fixed map expresses $1$.
• Figure 4: The sheaf of structures $\mathcal{H}^x$ with initial $0$-cochain $\rho(0)$, corresponding to the example in Figure \ref{['fig:stubborn-expressions-example']}. Vertex stalks are direct sums of Hom spaces; the stalk at $v_4$ (red) is fixed. The $0$-cochains $\rho_v$ are tuples of matrices corresponding to restriction maps in the original discourse sheaf $\mathcal{F}$. The restriction maps in $\mathcal{H}^x$ are evaluation maps at the fixed opinions $x$: the notation $\cdot(x_v)0$ extracts the first component and right-multiplies by $x_v$, while $\cdot0(x_v)$ extracts the second. Note that $x_{v_3} = 0$, so the restriction maps from $v_3$ annihilate any $0$-cochain, creating an unavoidable obstruction to consensus on edge $e_{34}$.
• Figure 5: The coupled dynamics of opinions and communication structures. Left: The opinion sheaf $\mathcal{F}^{\rho}$ over a single edge, with opinions $x_u \in \mathcal{F}(u)$ and $x_v \in \mathcal{F}(v)$ evolving under the sheaf Laplacian $L_{\mathcal{F}^{\rho}}$. Right: The structure sheaf $\mathcal{H}^{x}$ whose vertex stalks are the spaces $\mathrm{Hom}(\mathcal{F}(u), \mathcal{F}(e))$ and $\mathrm{Hom}(\mathcal{F}(v), \mathcal{F}(e))$; when stalks are finite-dimensional real vector spaces, these are simply spaces of matrices of compatible dimensions. The restriction maps $\mathrm{ev}_{x_u}$ and $\mathrm{ev}_{x_v}$ are evaluation maps that apply each matrix to the current opinion vector, i.e., right multiplication $\cdot \, x_u$ and $\cdot \, x_v$. The two dynamics are coupled: $L_{\mathcal{F}^{\rho}}$ depends on the current structure $\rho$, while $L_{\mathcal{H}^{x}}$ depends on the current opinions $x$.

Theorems & Definitions (44)

• Definition 2.1
• Theorem 2.2: Hansen and Ghrist hansen2020
• Definition 3.1: Sheaf of Free Opinions
• Theorem 3.2: Convergence to Equilibrium
• proof
• Remark 3.3: Relation to classical stubborn agents
• Theorem 4.1: Partial learning: convergence to the nearest consistent sheaf
• proof
• Remark 4.2: Relation to classical learning to lie
• Theorem 4.3: Regularized partial learning