Filtering amplitude dependence of correlation dynamics in complex systems: application to the cryptocurrency market

TL;DR

This work addresses how evolving cross-correlations in complex systems depend on fluctuation amplitude, using the cryptocurrency market as a testbed. It extends multifractal detrended cross-correlation analysis with $q$-dependent MSTs to quantify amplitude-filtered network structures from 1-minute returns of 140 assets on Binance, over 2021-01 to 2024-09, employing rolling windows and comparing $q=1$ vs $q=4$. The study uncovers a regime shift around May 2022, with BTC losing central hub status and the market becoming more decentralized; medium-scale fluctuations show stronger cross-correlations than large fluctuations, and removing the market factor alters but does not erase the core topology, highlighting distinct fluctuation-dependent correlation patterns. Practically, the $q$MST framework offers a flexible tool for pipeline-style portfolio optimization and risk management by tailoring strategies to fluctuation-specific correlation structures, with potential applicability to biology, social systems, and other complex networks.

Abstract

Based on the cryptocurrency market dynamics, this study presents a general methodology for analyzing evolving correlation structures in complex systems using the $q$-dependent detrended cross-correlation coefficient ρ(q,s). By extending traditional metrics, this approach captures correlations at varying fluctuation amplitudes and time scales. The method employs $q$-dependent minimum spanning trees ($q$MSTs) to visualize evolving network structures. Using minute-by-minute exchange rate data for 140 cryptocurrencies on Binance (Jan 2021-Oct 2024), a rolling window analysis reveals significant shifts in $q$MSTs, notably around April 2022 during the Terra/Luna crash. Initially centralized around Bitcoin (BTC), the network later decentralized, with Ethereum (ETH) and others gaining prominence. Spectral analysis confirms BTC's declining dominance and increased diversification among assets. A key finding is that medium-scale fluctuations exhibit stronger correlations than large-scale ones, with $q$MSTs based on the latter being more decentralized. Properly exploiting such facts may offer the possibility of a more flexible optimal portfolio construction. Distance metrics highlight that major disruptions amplify correlation differences, leading to fully decentralized structures during crashes. These results demonstrate $q$MSTs' effectiveness in uncovering fluctuation-dependent correlations, with potential applications beyond finance, including biology, social and other complex systems.

Figures (13)

• Figure 1: Evolution of the cumulative log-returns $\hat{R}(t)$ of the 140 cryptocurrencies over the time period from Jan 1, 2021 to Sep 30, 2024. The colors of two of the most liquid cryptocurrencies and a few other distinguished ones are indicated explicitly. The bulk of the cryptocurrencies is shown in the background (grey lines).
• Figure 2: Time evolution of the network characteristics of the $q$MSTs created from a distance matrix ${\bf D}(q=1,s=10)$: (a) node degree $k_j$ (cryptocurrencies that had the highest multiplicity in a given window were indicated), (b) average path length $\langle L \rangle$ and spectral characteristics of the $q$-dependent detrended correlation matrix ${\bf C}(q=1,s=10)$: (c) the largest eigenvalue $\lambda_1$, (d) the squared expansion coefficients of the eigenvector ${ v}^{2}_{1,j}$ associated with $\lambda_1$ for $j$=BTC, ETH, MANA, LINK, and SAND (e) the Shannon entropy H(${\bf v}^{2}_{1}$) of the squared eigenvector components. Rolling window of length 7 days shifted by 1 day was applied.
• Figure 3: Sample $q$MSTs calculated for ($q=1$ and $s=10$) in the rolling windows ending on: (a) Apr 25, 2022 and (b) May 18, 2022. Node size is proportional to the average volume in the analyzed period, while edge thickness reflects the strength of the cross-correlations. Colors represent market sectors after Digital Asset Classification Standard (DACS), created by CoinDesk coindesk: currency (orange), smart contract platform (violet), computing (cyan), DeFi (green), and culture & entertainment (dark green).
• Figure 4: Time evolution of the network characteristics of the $q$MSTs created from a distance matrix ${\bf D}(q=4,s=10)$: (a) node degree $k_j$ (cryptocurrencies that had the largest multiplicity in a given window were indicated) and spectral characteristics of the $q$-dependent detrended correlation matrix ${\bf C}(q=4,s=10)$: (b) the squared expansion coefficients of the eigenvector ${\bf v}^{2}_{1,j}$ associated with $\lambda_1$ for $j$=BTC, ETH, SAND, MANA, and LINK.
• Figure 5: Time evolution of the network characteristics of the $q$MSTs created from a distance matrix ${\bf D}(q=1,s=10)$ and ${\bf D}(q=4,s=10)$: (a) max node degree $k_{\textrm{max}}$, (b) average path length $\langle L \rangle$, (c) $d_{\textrm{rp}1}$, and (d) $d_{\textrm{DC}0}$ between $q=1$ and $q=4$ MST. The spectral characteristics of the $q$-dependent detrended correlation matrix ${\bf C}(q=1,s=10)$ and ${\bf C}(q=4,s=10)$: (e) the largest eigenvalue $\lambda_1$, (f) the highest squared expansion coefficients of the eigenvector ${\bf v}^{2}_{1,\textrm{max}}$ associated with $\lambda_1$, and (g) the Shannon entropy H(${\bf v}^{2}_{1}$) of the squared eigenvector components. Rolling window of length 7 days shifted by 1 day was applied. Periods with large intraday drops are marked with Roman numerals and dotted lines.