A class of networks are those with both positive and negative links. In this manuscript, we studied the interplay between positive and negative ties on mesoscopic level of these networks, i.e., their community structure. A community is considered as a tightly interconnected group of actors; therefore, it does not borrow any assumption from balance theory and merely uses the well-known assumption in the community detection literature. We found that if one detects the communities based on only positive relations (by ignoring the negative ones), the majority of negative relations are already placed between the communities. In other words, negative ties do not have a major role in community formation of signed networks. Moreover, regarding the internal negative ties, we proved that most unbalanced communities are maximally balanced, and hence they cannot be partitioned into k nonempty sub-clusters with higher balancedness (k ≥ 2). Furthermore, we showed that although the mediator triad + + − (hostile-mediator-hostile) is underrepresented, it constitutes a considerable portion of triadic relations among communities. Hence, mediator triads should not be ignored by community detection and clustering algorithms. As a result, if one uses a clustering algorithm that operates merely based on social balance, mesoscopic structure of signed networks significantly remains hidden.

Extracting community structure of complex network systems has many applications from engineering to biology and social sciences. There exist many algorithms to discover community structure of networks. However, it has been significantly under-explored for networks with positive and negative links as compared to unsigned ones. Trying to fill this gap, we measured the quality of partitions by introducing a Map Equation for signed networks. It is based on the assumption that negative relations weaken positive flow from a node towards a community, and thus, external (internal) negative ties increase the probability of staying inside (escaping from) a community. We further extended the Constant Potts Model, providing a map spectrum for signed networks. Accordingly, a partition is selected through balancing between abridgment and expatiation of a signed network. Most importantly, multi-scale spectrum of signed networks revealed how informative are negative ties in different scales, and quantified the topological placement of negative ties between dense positive ones. Moreover, an inconsistency was found in the signed Modularity: as the number of negative ties increases, the density of positive ties is neglected more. These results shed lights on the community structure of signed networks.

No abstract

No abstract