2-Edge Connected Balanced Subgraphs for Correlation Clustering Problem

Abstract : A signed graph, whose links are labeled as positive (+) and negative (−), is considered structurally balanced if it can be partitioned into a number of clusters, such that positive (negative) links are located inside (in-between) the clusters. Due to the imbalanced nature of real-world networks, various measures have been defined to quantify the amount of imbalance. Such measures are expressed relatively to a graph partition, so processing the graph balance amounts to identifying the partition corresponding to the lowest imbalance measure. A well-known measure among them corresponds to the definition of the Correlation Clustering (CC) problem, and it consists in counting the numbers of misplaced (w.r.t. structural balance definition) links. One issue of the CC problem is that it is solely based on misplaced links, and it may not always reflect the real situations. One extreme but plausible case is the network instance of two positive groups where they are connected by the same number of positive and negative links, and this produces two possible optimal solutions. To overcome these situations and those being in similar fashion one needs to extend the CC problem by adding additional topological constraints. The topology of the network is an important aspect in graph theory, and it is nearly always present in practical situations (e.g. in social dynamics). Indeed, topology constitutes the base of the problems in social networks (e.g. community detection), since they rely solely on this type of criteria. In this work, we are inspired by the studies ensuring network robustness (e.g. single failure in telecommunication, robustness to the damage by mutation or viral infection in biological networks), and we propose to extend the CC problem with 2 positive edge connectivity requirement at cluster level. Hence, the network topology provides for at least two diverse positive paths between each pair of nodes in the same cluster. In literature, although 2-edge connectivity constraint is well studied (for nonsigned networks), to the best of our knowledge, the only work considering 2-edge connectivity constraint for the CC problem is on planar graphs (i.e. graph that can be drawn on the plane without intersection of its edges), but not for the general case.
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Nejat Arinik, Rosa Figueiredo, Vincent Labatut. 2-Edge Connected Balanced Subgraphs for Correlation Clustering Problem. 21ème Congrès Annuel de la Société Française de Recherche Opérationnelle et d’Aide à la Décision (ROADEF), Société Française de Recherche Opérationnelle et d’Aide à la Décision, Feb 2020, Montpellier, France. ⟨hal-02428305⟩

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