Pattern formation from non-local stifling

Establish that, under the homogeneous hypotheses for the Daley–Kendall opinion-propagation system with non-local stifling, there exist parameter values a, b, and R_β for which the limiting ignorant populations are spatially heterogeneous, specifically satisfying limsup_{n→∞} S_n^∞ > liminf_{n→∞} S_n^∞.

Background

The paper studies a non-local stifling mechanism in which spreaders at node n respond not only to opinion adoption at n but also to adoption within graph distance R_β. Numerical simulations on a path graph and a Watts–Strogatz graph suggest that sufficiently long-range stifling can create persistent spatial patterns or alternating regions with different final adoption levels.

The conjecture formalizes this simulation-based observation by asserting the existence of parameters that make the limiting ignorant population non-homogeneous. The convergence of the nodewise limits is established, but the existence of a parameter regime with strict asymptotic spatial oscillation remains unresolved.