Unimodularity sufficiency for realizability of local limit objects

Determine whether unimodularity (involution invariance) of a limit object in the local convergence framework for sparse random graphs is sufficient to guarantee the existence of a random graph sequence whose local limit equals that object. Specifically, establish if every unimodular limit object can be realized as the local limit of some random graph.

Background

In the study of local convergence for sparse random graphs, unimodularity—equivalently involution invariance—is a central property of limit objects, capturing a form of statistical symmetry. The paper notes that while unimodularity is necessary, it is unknown in general whether this property is sufficient to guarantee the existence of a random graph sequence that converges locally to a given unimodular object.

The authors highlight that, for certain classes of limit objects such as specific Galton–Watson trees, sufficiency is known. However, the general question remains unresolved, marking a fundamental gap in the theory of sparse graph limits.