Multiplicity and diversity: analysing the optimal solution space of the correlation clustering problem on complete signed graphs

• Published in: Journal of complex networks Vol. 8; no. 6
• Main Authors: Arınık, Nejat, Figueiredo, Rosa, Labatut, Vincent
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01-12-2020
• Subjects: Computer Science; Operations Research; Social and Information Networks; signed graph; solution space; structural balance; multiple solutions; graph partitioning; correlation clustering; Signed Graph; Correlation Clustering; Multiple Solutions; Solution Space; Complete Graph; Structural Balance; Graph Partitioning
• ISSN: 2051-1310; 2051-1329
• DOI: 10.1093/comnet/cnaa025

Abstract
In order to study real-world systems, many applied works model them through signed graphs, that is, graphs whose edges are labelled 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 (respectively negative) edges are located inside (respectively 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.