Figure 14

Intricate dynamics of sociotechnical time series. Sociotechnical time series can display intricate dynamics and extended periods of anomalous behavior. The red curve shows the time series of the ranks down from top of the word “bling” on Twitter. Until 2015/10/31, the time series presents as random fluctuation about a steady trend that is nearly indistinguishable from zero. However, the series then displays a large fluctuation, increases rapidly, and then decays slowly after a sharp peak. The underlying mechanism for these dynamics was the release of a popular song titled “Hotline Bling” by a musician known as “Drake”. Returns \(\Delta r_{t} = r_{t + 1} - r _{t}\) are calculated and their histogram is displayed in panel (C). To demonstrate the qualitative difference of the “bling” time series from other time series with an identical returns distribution, elements of the symmetric group \(\sigma _{i} \in \mathcal{S}_{T}\) are applied to the returns of the original series, \(\Delta r_{t} \mapsto \Delta r_{\sigma _{i}t}\), and the resultant noise is integrated and plotted as \(r_{\sigma _{i} t} = \sum_{t' \leq t}\Delta r_{\sigma _{i}t}\). The bottom-left panel (C) displays time-decoupled probability distributions of the returns of the plotted time series. The distributions of \(\Delta r_{i}\) and \(\sigma \Delta r_{i}\) are identical, as they should be, but the integrated series have entirely different spectral behavior and dynamic ranges. Panels (D)–(G) display the discrete shocklet transform of the original series and the random walks \(\sum_{t'\leq t} \Delta r _{\sigma _{i} t}\), showing the responsiveness of the DST to nonstationary local dynamics and its insensitivity to dynamic range. The right-most column of panels (H)–(K) displays the discrete wavelet transform of the original series demonstrating its comparatively less-sensitive nature to local anomalous dynamics