Fig. 2
Detecting abrupt transitions with communities in networks of recurrence probabilities. The time series of PDFs \(\left\{ {\varrho _t^X} \right\}_{t = 1}^N\) for a synthetically generated noisy sinusoid (color map in (a)) and its mean (d). Three transitions are imposed: (1) a sudden jump at T = 200, (2) a linear decrease between T = 400 and T = 450, and (3) a change in the PDF to bimodality at T = 675. The probability of recurrence matrix \({{\hat {\bf A}}}\) in (b), estimated from the densities in (a), shows the modular structure resulting from the imposed transitions. The recurrence matrix R estimated from the mean time series (e) captures only the first two transitions. We detect the timing of the transitions by moving a sliding window (white box in (b)) of 100 time points and estimating the p-value (c, f) for a two-community structure under the null hypothesis of a random network. Statistically significant p-values (plus signs in (a)) are determined at a level α = 0.05, and after accounting for multiple comparisons using Holm’s method with the Dunn–Šidák correction factor (see Methods). In (f), the third transition is not detected and the first two are much more coarsely dated than in (c)
