Relationship between node centrality in the Gremban expansion and the original signed network

Determine the relationship between node centrality values computed on the Gremban expansion of a signed graph and node centrality values in the original signed network, including how centrality scores and rankings project between the expanded unsigned graph and the original signed graph, to clarify whether and how centrality is preserved or transformed by the expansion and projection maps.

Background

The paper develops the Gremban expansion, which lifts a signed graph to an unsigned double-cover graph exhibiting an involutive symmetry between positive and negative node polarities. This expansion enables the use of standard unsigned spectral and combinatorial tools for analyzing signed networks, including community and faction detection, and provides clear correspondences for certain structures (e.g., cut-sets and frustration sets) under symmetric partitions.

While the work establishes correspondences for mesoscale structures and dynamics (e.g., spectral clustering, random walks, diffusion), it does not establish how node-level centrality measures translate between the expanded graph and the original signed network. Because the Gremban expansion duplicates nodes and encodes signs structurally, understanding how centrality rankings or magnitudes on the expansion relate to centrality on the original signed graph is nontrivial and left unresolved.