Community detection in signed networks: The role of negative ties in different scales

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