Spectral Analysis of k-Balanced Signed Graphs

Previous studies on social networks are often focused on networks with only positive relations between individual nodes. As a significant extension, we conduct the spectral analysis on graphs with both positive and negative edges. Specifically, we investigate the impacts of introducing negative edges and examine patterns in the spectral space of the graph's adjacency matrix. Our theoretical results show that communities in a k-balanced signed graph are distinguishable in the spectral space of its signed adjacency matrix even if connections between communities are dense. This is quite different from recent findings on unsigned graphs, where communities tend to mix together in the spectral space when connections between communities increase. We further conduct theoretical studies based on graph perturbation to examine spectral patterns of general unbalanced signed graphs. We illustrate our theoretical findings with various empirical evaluations.