Examining spectral space of complex networks with positive and negative links

Abstract

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 become dense. 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 on both synthetic data and real world Correlates of War data.

Modularized convex nonnegative matrix factorization for community detection in signed and unsigned networks

“…[27]: The network is about the relations among the ten parties in Slovene parliament, 1994 [27]. It has two communities: (1,3,6,8,9) and (2, 4, 5, 7, 10) [28]. The weights of the links in the network were estimated by experts on parliament activities meaning the relations among the parties.…”

Bayesian Approach: An Alternative to Periodogram and Time Axes Estimation for Known and Unknown White Noise

“…[5] gave a clear picture of how spectral time series of multispectral and periodogram recognition schemes in the contexts of image acquisition, iris segmentation, texture analysis, and matching and performance evaluation while [6] thoroughly dealt with Fourier analysis on graphs with both positive and negative edges; [6] investigated the impacts of introducing negative edges and examine patterns in the spectral space of the graphs' adjacency matrix. Their theoretical results [5] and [6] showed 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 with an illustration empirical evaluation on both synthetic data and real life data.…”

“…Their theoretical results [5] and [6] showed 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 with an illustration empirical evaluation on both synthetic data and real life data. [7] Also maintained that the Wigner quasi-distribution plays an alternative role in both time-frequency analysis and quantum mechanics the white noise instead of the conventional Gaussian distribution been use in both the …”

Community detection in signed networks based on discrete-time model

“…Commonly, the social relationships between individuals have positive relationships and negative relationships. [4] It will be regarded as a positive link when two individuals have a friendly relationship, however, it will be regarded as a negative link when two individuals have a hostile relationship. Such networks are called signed networks.…”