Nonlinear dynamics of river runoff elucidated by horizontal visibility graphs

Abstract

Horizontal Visibility Graphs (HVGs) are a recently developed method to construct networks from time series. The values of the time series are considered as the nodes of the network and are linked to each other if there is no larger value between them, such as they can "see" each other. The network properties reflect the nonlinear dynamics of the time series. For some classes of stochastic processes and for periodic time series, analytical results can be obtained for network-derived quantities such as the degree distribution, the local clustering coefficient distribution, the mean path length, and others. HVGs have the potential to discern between deterministic-chaotic and correlated-stochastic time series. Here, we investigate the sensitivity of the HVG methodology to properties and pre-processing of real-world data, i.e., time series length, the presence of ties, and deseasonalization, using a set of around 150 runoff time series from managed rivers at daily resolution from Brazil with an average length of 65 years. We show that an application of HVGs on real-world time series requires a careful consideration of data pre-processing steps and analysis methodology before robust results and interpretations can be obtained. For example, one recent analysis of the degree distribution of runoff records reported pronounced sub-exponential "long-tailed" behavior of North American rivers, whereas another study of South American rivers showed hyper-exponential "short-tailed" behavior resembling correlated noise. We demonstrate, using the dataset of Brazilian rivers, that these apparently contradictory results can be reconciled by minor differences in data-preprocessing (here: small differences in subtracting the seasonal cycle). Hence, data-preprocessing that is conventional in hydrology ("deseasonalization") changes long-term correlations and the overall runoff dynamics substantially, and we present empirical consequences and extensive simulations to investigate these issues from a HVG methodological perspective. After carefully accounting for these methodological aspects, the HVG analysis reveals that the river runoff dataset shows indeed complex behavior that appears to stem from a superposition of short-term correlated noise and "long-tailed behaviour," i.e., highly connected nodes. Moreover, the construction of a dam along a river tends to increase short-term correlations in runoff series. In summary, the present study illustrates the (often substantial) effects of methodological and data-preprocessing choices for the interpretation of river runoff dynamics in the HVG framework and its general applicability for real-world time series.

Insights on Streamflow Predictability Across Scales Using Horizontal Visibility Graph Based Networks

“…(normalized) the time series using the mean and standard deviation of flows for each day of the year (e.g., Braga et al, 2016;Serinaldi and Kilsby, 2016;Lange et al, 2018), a typical procedure for such hydrologic analyses. We discuss later the impact of this transformation on inferences derived from the networks.…”

“…A rationale behind using normalized streamflow time series is to avoid potential seasonal trends in streamflow (see Braga et al, 2016;Serinaldi and Kilsby, 2016;Lange et al, 2018). Here, we performed a similar analysis for λ and GC using raw streamflow time series.…”

“…They demonstrated the presence of correlated stochastic structure in the streamflow dynamics through degree distribution and GC. Further, Lange et al (2018) investigated the sensitivity of the HVG methodology to streamflow time series pre-processing properties such as time series length, presence of ties in the data, and effects of seasonality, using around 150 time series from managed rivers in Brazil at a daily time scale. They showed that data pre-processing can result in contradictory results and thus should be used with caution.…”

Analysis of Air Mean Temperature Anomalies by Using Horizontal Visibility Graphs

“…Before employing the HVG algorithm, seasonal behavior of signals has been removed by computing the mean temperature anomalies. These new time series are obtained by subtracting the average value for each calendar day over the whole period from the original time series ( ) and normalizing by their standard error ( ), i.e., ′ = ( − )⁄ , for = 1 … 366 day [25]. In Figure 2a,b, two examples of time series have been depicted.…”

Introduction to Focus Issue: Nonlinear dynamics of non-equilibrium complex systems

“…Also, by means of some simulations on a real subway network, how to use a personalization vector suitable for biasing the PageRank is shown with the aim of determining the segments of the subway network with the highest number of passengers depending on the time zone under consideration. Lange et al (2018) investigate the dynamics of water flow, given as time series of river runoff from long-term measurement stations (up to 85 years of data) in Brazil. The time series are analyzed using "Horizontal Visibility Graphs" (HVG).…”