Asymptotic homogeneity under local stifling

Prove that, under the homogeneity hypotheses for the Daley–Kendall opinion-propagation system on a graph, with purely local stifling and finite initial support of the opinion, there exists a constant S^∞>0 independent of the node such that the limiting ignorant population S_n^∞ converges to S^∞ as the graph distance d(n) tends to infinity.

Background

The paper defines the model as a network generalization of the Daley–Kendall rumor model, with S_n, I_n, and R_n denoting ignorant, spreading, and stifler populations at node n. In the homogeneous setting, the initial ignorant population is constant, transmission rates are normalized uniformly over neighboring nodes, and stifling rates are normalized uniformly over nodes within a prescribed range.

For purely local stifling (R_β=0), the paper proves convergence of S_n(t) to a node-dependent limit S_n∞ and obtains uniform bounds, but it does not establish that the limiting adoption level becomes asymptotically identical at nodes far from the initially seeded region. The stated conjecture asks for precisely this asymptotic spatial homogeneity.

References

If these homogeneity hypotheses hold true and if the stifling is local (that is, R_\beta=0), then we conjecture that there is S\infty>0 (independent of n) such that $$ S_n\infty \underset{d(n)\to+\infty}{\longrightarrow}S\infty. $$ We leave the proof of this fact as an open question for future works.

— Propagation of opinions on a network. Convergence, spreading speed and patterns in the Daley-Kendall model  (2609.26543 - Ducasse, 22 Sep 2026) in Remark ‘Homogeneous case,’ Remark \ref{conj hom}, immediately following Theorem \ref{th speed}