Core community structure recovery and phase transition detection in temporally evolving networks

Community detection in time series networks represents a timely and significant research topic due to its applications in a broad range of scientific fields, including biology, social sciences and engineering. In this work, we introduce methodology to address this problem, based on a decomposition of the network adjacency matrices into low-rank components that capture the community structure and sparse & dense noise perturbation components. It is further assumed that the low-rank structure exhibits sharp changes (phase transitions) at certain epochs that our methodology successfully detects and identifies. The latter is achieved by averaging the low-rank component over time windows, which in turn enables us to precisely select the correct rank and monitor its evolution over time and thus identify the phase transition epochs. The methodology is illustrated on both synthetic networks generated by various network formation models, as well as the Kuramoto model of coupled oscillators and on real data reflecting the US Senate's voting record from 1979-2014. In the latter application, we identify that party polarization exhibited a sharp change and increased after 1993, a finding broadly concordant with the political science literature on the subject.