Multiplicity and Diversity: Analyzing the Optimal Solution Space of the Correlation Clustering Problem on Complete Signed Graphs

TL;DR

This paper investigates the solution space of the Correlation Clustering problem on signed graphs, revealing the existence of multiple diverse optimal partitions which can offer different insights into the underlying system.

Contribution

It empirically demonstrates the multiplicity and diversity of optimal solutions in correlation clustering, highlighting the importance of exploring multiple solutions for comprehensive analysis.

Findings

Multiple optimal partitions can exist for the same signed graph.

Some optimal solutions are significantly different, providing diverse system perspectives.

Studying multiple solutions is crucial for a thorough understanding of the system.

Abstract

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.