  In order to study real-world systems, many applied works model them through
signed graphs, i.e. graphs whose edges are labeled as either positive or
negative. Such a graph is considered as structurally balanced when it can be
partitioned into a number of modules, such that positive (resp. negative) edges
are located inside (resp. in-between) the modules. When it is not the case,
authors look for the closest partition to such balance, a problem called
Correlation Clustering (CC). Due to the complexity of the CC problem, the
standard approach is to find a single optimal partition and stick to it, even
if other optimal or high scoring solutions possibly exist. In this work, we
study the space of optimal solutions of the CC problem, on a collection of
synthetic complete graphs. We show empirically that under certain conditions,
there can be many optimal partitions of a signed graph. Some of these are very
different and thus provide distinct perspectives on the system, as illustrated
on a small real-world graph. This is an important result, as it implies that
one may have to find several, if not all, optimal solutions of the CC problem,
in order to properly study the considered system.
