# Exploring Overall and Component Complexities via Relative Complexity Change and Interacting Complexity Amplitudes in the Kolmogorov Plane: A Case Study of U.S. Rivers

## TL;DR

This paper uses Kolmogorov complexity to study how environmental factors contribute to the complexity of U.S. river systems over time.

## Contribution

The novel approach combines KC metrics with relative complexity change to analyze streamflow complexity patterns.

## Key findings

- Normalized KC spectra revealed interactive complexity amplitudes in U.S. rivers.
- Relative complexity change (RCC) showed distinct intervals of complexity patterns.
- Environmental variables like temperature and precipitation influence streamflow complexity.

## Abstract

One of the most challenging tasks in studying streamflow is quantifying how the complexities of environmental and dynamic parameters contribute to the overall system complexity. To address this, we employed Kolmogorov complexity (KC) metrics, specifically the Kolmogorov complexity spectrum (KC spectrum) and the Kolmogorov complexity plane (KC plane). These measures were applied to monthly streamflow time series averaged across 1879 gauge stations on U.S. rivers over the period 1950–2015. The variables analyzed included streamflow as a complex physical system, along with its key components: temperature, precipitation, and the Lyapunov exponent (LEX), which represents river dynamics. Using these metrics, we calculated normalized KC spectra for each position within the KC plane, visualizing interactive master amplitudes alongside individual amplitudes on overlapping two-dimensional planes. We further computed the relative change in complexities (RCC) of the normalized master and individual components within the KC plane, ranging from 0 to 1 in defined intervals. Based on these results, we analyzed and discussed the complexity patterns of U.S. rivers corresponding to each interval of normalized amplitudes.