Modularity, balance, and frustration in student social networks: The role of negative relationships in communities.

Signed networks provide information to study the structure and composition of relationships (positive and negative) among individuals in a complex system. Individuals, through different criteria, form groups or organizations called communities. Community structures are one of the important properties of social networks. In this work, we aim to analyze the perturbation of negative relationships in communities. We developed a methodology to obtain and analyze the optimal community partitions in nine school networks in the state of Yucatán, México. We implemented a technique based on the social balance theory in signed networks to complete negative missing links and further applied two methods of community detection: Newman's and Louvain's algorithms. We obtain values close to Dunbar's ratio for both types of relationships, positive and negative. The concepts of balance and frustration were analyzed, and modularity was used to measure the perturbation of negative relationships in communities. We observe differences among communities of different academic degrees. Elementary school communities are unstable, i.e. significantly perturbed by negative relationships, in secondary school communities are semi-stable, and in high school and the university the communities are stable. The analyzes indicate that a greater number of negative links in the networks does not necessarily imply higher instability in the communities, but other social factors are also involved.