Fig. 3: Geometry of primal and dual community boundaries.

a Decomposition of a graph into two primal communities. Some edges belong to neither of the two communities, but provide a weak connection between the communities. These edges constitute the cut-set S. b A graph (black solid lines) and its dual (gray dashed lines), which is decomposed into two communities \({V}_{1,2}^{*}\). Some primal edges, colored in green, belong to faces from both communities. These primal edges, together with their terminal nodes, constitute the cut-path p.