Modeling dynamic social networks using spectral embedding

Dynamic social network analysis aims to understand the structures in networks as they evolve, as nodes appear and disappear, and as edge weights change. Working directly with a social network graph is difficult, and it has become standard to use spectral techniques that embed a graph in a geometry. Analysis can then be done in the geometry where distance approximates dissimilarity. Recently, spectral techniques have been extended to model directed graphs; we build on these techniques to model directed graphs that change over time. The snapshots of the social network at each time period are bound together into a single graph in a way that keeps structures aligned over time, and this global graph is then spectrally embedded. The similarities among a set of nodes can be tracked over time, so that changing relationships and clusters can be seen; and the concept of the trajectory of a node across time also becomes meaningful. We illustrate how these approaches can be used to understand the changing social network of the Caviar drug-trafficking network under both internal dynamics and response to law-enforcement actions.