The construction of a meaningful graph plays a crucial role in the success of many graph-based representations and algorithms for handling structured data, especially in the emerging field of graph signal processing. However, a meaningful graph is not always readily available from the data, nor easy to define depending on the application domain. In particular, it is often desirable in graph signal processing applications that a graph is chosen such that the data admit certain regularity or smoothness on the graph. In this paper, we address the problem of learning graph Laplacians, which is equivalent to learning graph topologies, such that the input data form graph signals with smooth variations on the resulting topology. To this end, we adopt a factor analysis model for the graph signals and impose a Gaussian probabilistic prior on the latent variables that control these signals. We show that the Gaussian prior leads to an efficient representation that favors the smoothness property of the graph signals. We then propose an algorithm for learning graphs that enforces such property and is based on minimizing the variations of the signals on the learned graph. Experiments on both synthetic and real world data demonstrate that the proposed graph learning framework can efficiently infer meaningful graph topologies from signal observations under the smoothness prior.

The construction of a meaningful graph topology plays a crucial role in the effective representation, processing, analysis, and visualization of structured data. When a natural choice of the graph is not readily available from the data sets, it is thus desirable to infer or learn a graph topology from the data. In this article, we survey solutions to the problem of graph learning, including classical viewpoints from statistics and physics, and more recent approaches that adopt a graph signal processing (GSP) perspective. We further emphasize the conceptual similarities and differences between classical and GSP-based graph-inference methods and highlight the potential advantage of the latter in a number of theoretical and practical scenarios. We conclude with several open issues and challenges that are keys to the design of future signal processing and machine-learning algorithms for learning graphs from data.

The effective representation, processing, analysis, and visualization of large-scale structured data, especially those related to complex domains, such as networks and graphs, are one of the key questions in modern machine learning. Graph signal processing (GSP), a vibrant branch of signal processing models and algorithms that aims at handling data supported on graphs, opens new paths of research to address this challenge. In this article, we review a few important contributions made by GSP concepts and tools, such as graph filters and transforms, to the development of novel machine learning algorithms. In particular, our discussion focuses on the following three aspects: exploiting data structure and relational priors, improving data and computational efficiency, and enhancing model interpretability. Furthermore, we provide new perspectives on the future development of GSP techniques that may serve as a bridge between applied mathematics and signal processing on one side and machine learning and network science on the other. Cross-fertilization across these different disciplines may help unlock the numerous challenges of complex data analysis in the modern age.

Urban income segregation is a widespread phenomenon that challenges societies across the globe. Classical studies on segregation have largely focused on the geographic distribution of residential neighborhoods rather than on patterns of social behaviors and interactions. In this study, we analyze segregation in economic and social interactions by observing credit card transactions and Twitter mentions among thousands of individuals in three culturally different metropolitan areas. We show that segregated interaction is amplified relative to the expected effects of geographic segregation in terms of both purchase activity and online communication. Furthermore, we find that segregation increases with difference in socio-economic status but is asymmetric for purchase activity, i.e., the amount of interaction from poorer to wealthier neighborhoods is larger than vice versa. Our results provide novel insights into the understanding of behavioral segregation in human interactions with significant socio-political and economic implications.

Social behaviours emerge from the exchange of information among individuals—constrained by and reciprocally influencing the structure of information flows. The Internet radically transformed communication by democratizing broadcast capabilities and enabling easy and borderless formation of new acquaintances. However, actual information flows are heterogeneous and confined to self-organized echo-chambers. Of central importance to the future of society is understanding how existing physical segregation affects online social fragmentation. Here, we show that the virtual space is a reflection of the geographical space where physical interactions and proximity-based social learning are the main transmitters of ideas. We show that online interactions are segregated by income just as physical interactions are, and that physical separation reflects polarized behaviours beyond culture or politics. Our analysis is consistent with theoretical concepts suggesting polarization is associated with social exposure that reinforces within-group homogenization and between-group differentiation, and they together promote social fragmentation in mirrored physical and virtual spaces.