A complex network approach to characterize clustering of events in irregular time series

Abstract

In complex systems, events occur at irregular intervals that inherently encode the underlying dynamics of the system. Analyzing the temporal clustering of these events reveals critical insights into the non-random patterns and the temporal evolution. Existing techniques can effectively quantify the overall clustering tendency of events using global statistical measures. However, these macroscopic approaches leave a critical gap, as they do not attempt to investigate the dynamics of individual clusters. Analyzing individual clusters is essential, as it helps comprehend the local interactions that actively drive the system dynamics, which may be obscured by global averaging, while simultaneously revealing the time scales involved. To address these limitations, we propose a complex network-based framework for analyzing clustering of events occurring at irregular intervals. The framework establishes connections using arrival times, transforming the time series into a network. Network properties are then used to quantify the clustering. Further, a community detection algorithm is used to identify individual clusters in time series. We illustrate the method by applying it to standard arrival processes, such as the Poisson process and the Markov-modulated Poisson process. To further demonstrate its scope, we apply the method to two diverse systems: the time series of droplet arrivals in turbulent flows and the R-R intervals in electrocardiogram (ECG) signals.

Figures (17)

• Figure 1: Schematic of construction of network from irregular arrival time series. (a) Irregular arrival time series with arrivals represented as vertical lines. (b) Section of the time series showing network construction for arrival $i$ with time window $\tau$ forward and backward. (c) Section of the time series represented as nodes and links after completing the network construction.
• Figure 2: Time series generated for different arrival processes. The vertical lines represent arrivals in a time series. A two-second section with 200 arrivals in each time series is shown. Arrivals are equally spaced for the regular arrival process. The periods of inactivity and arrivals are much closer in MMPP, than in Poisson's arrival process.
• Figure 3: Comparison of global clustering measures average node strength $S_{avg}$ and fishing statistic $F$. Both measures are zero for regular intervals. For Poisson arrival process, $F$ is zero because it measures deviations from the Poisson arrival processes, while the $S_{avg}$ provides the absolute value of clustering. For MMPP, both measures accurately estimate the higher levels of clustering observed.
• Figure 4: Node strength distribution for (a) regular arrival process, (b) Poisson’s arrival process, and (c) MMPP. In the case of regular arrivals, all nodes have the same node strength. For the Poisson arrival process and MMPP, the distribution is not concentrated at a single value, indicating variability in node strengths, with MMPP showing comparatively higher values.
• Figure 5: The network structure and the adjacency matrix for each arrival process. (a) The networks are drawn using Fruchterman and Reingold algorithmfruchterman1991graph in Gephi visualization software.bastian2009gephi Visualization of the complex networks show that MMPP has locations of packed node structures (nodes with dark blue color), and the network is divided into separate groups (groups are separated by red dashed borders). The Poisson's arrivals have a few packed structures; however, there are no separate groups. The regular arrival process also has no groups and is seen as a connected network. (b) The adjacency matrix with detected communities marked by the orange boxes is plotted for each process. We observe no communities for regular arrivals. For Poisson’s arrival process, there is an emergence of a few small communities, whereas larger communities are seen for MMPP.