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Time-dependent weighted directed networks of cryptocurrency interaction from high-frequency returns

Using high-frequency data from 2020 to 2025, this study constructs time-dependent weighted directed networks based on Granger causality to reveal a dynamically evolving hierarchy in cryptocurrency markets where Ethereum consistently surpasses Bitcoin as the most influential asset, highlighting the ecosystem's competitive and non-stable nature.

Original authors: Shubhangam Shukla, Mahesh Peyyala, Abhijit Chakraborty

Published 2026-06-25

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CC BY 4.0

Original authors: Shubhangam Shukla, Mahesh Peyyala, Abhijit Chakraborty

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the cryptocurrency market not as a collection of isolated coins, but as a bustling, chaotic city where every coin is a person constantly talking to everyone else. Some people are loud and influential, shouting news that makes others react. Others are quiet listeners, or perhaps they only speak when the "loud" ones do.

This paper is like a detective story where the authors used a special kind of "microphone" to listen to the conversations between these coins over five years (2020–2025). Here is what they found, explained in simple terms:

1. The Setup: Listening to the Whispers

The researchers didn't just look at the price of a coin; they looked at how the price moved every single minute. They treated these price movements like a conversation. If Coin A's price jumps, does Coin B's price jump a moment later?

They used a statistical tool called Granger Causality. Think of this as a "predictive test." It asks: "If I know what happened to Coin A in the past, can I predict what Coin B will do better than if I just guessed?" If the answer is yes, they draw an arrow from A to B, saying, "A is influencing B."

2. The Noise: Wild Swings

First, they checked the "voice" of the coins. They found that price changes are wild and unpredictable. Most of the time, the coins whisper, but occasionally, they scream.

The Analogy: Imagine a calm lake that suddenly has massive, rare tsunamis. The paper found that these "tsunamis" (huge price jumps or drops) happen more often than you'd expect in a normal, calm world. This is a common trait in financial markets, but it's very pronounced here.

3. The Map: A Lopsided City

When they mapped out who influences whom, they didn't find a circle of equals. They found a very uneven hierarchy.

The Analogy: Imagine a city where 90% of the people are quiet neighbors, but a tiny group of 5% are megaphone-wielding celebrities. These few "celebrity coins" are responsible for almost all the noise and influence in the city.
The Finding: The network is "heterogeneous," meaning the influence is not shared equally. A small handful of coins drive the market dynamics.

4. The Ranking: The Shifting Throne

The most exciting part of the study is how the "leaders" changed over time. The authors ranked the coins based on how much they influenced others (their "out-strength").

Ethereum (ETH): This coin was the undisputed king for the entire five years. It consistently sat at the #1 spot. The authors suggest this is because Ethereum keeps evolving (like upgrading its engine to a new, more efficient system), keeping people confident in its future.
Bitcoin (XBT): Once the clear leader, Bitcoin's relative influence has been slowly fading. It's still a giant, but it's no longer the only one calling the shots.
The Churn: The top 5 spots were a revolving door. Coins like Ripple (XRP) started strong but fell out of favor (partly due to legal trouble with regulators). Newer coins like Solana (SOL) climbed the ranks quickly.
The Big Lesson: Unlike other complex systems (like the human brain or the internet) where the "super-stars" stay super-stars for decades, the crypto market is a highly competitive reality show. The top spot changes frequently, and no one is safe from being replaced.

5. The "Echo" Check: Is it Real Influence?

A skeptic might ask: "Maybe these coins aren't talking to each other; maybe they are all just reacting to the same big news (like a market crash) at the same time."

To test this, the researchers removed the "common background noise" (the general market mood) from their data and ran the test again.

The Result: The map looked almost exactly the same. This proves that the connections they found are real, direct conversations between specific coins, not just everyone reacting to the same weather.

Summary

This paper tells us that the cryptocurrency market is a dynamic, competitive ecosystem. It's not a stable club where the same leaders rule forever. Instead, it's a fast-moving arena where influence flows heavily from a few key players, but the lineup of those players changes constantly. Ethereum has taken the crown from Bitcoin in terms of influence, and the market remains a place of constant, heavy-tailed volatility where big surprises are always possible.

Technical Summary: Time-dependent weighted directed networks of cryptocurrency interaction from high-frequency returns

Problem and Motivation
Cryptocurrency markets operate as a complex, interacting system characterized by decentralized governance, continuous trading, and pronounced price fluctuations across multiple time scales. Unlike traditional financial markets, these assets are not isolated; they form a dynamic network where information and shocks propagate via shared sentiment, arbitrage, and external influences. While previous studies have examined cryptocurrency networks using transaction-level data or shorter time windows, there is a need to understand the temporal evolution of causal interactions using high-frequency data over extended periods. This study addresses the challenge of quantifying the flow of influence across assets and characterizing the structural stability (or lack thereof) of the cryptocurrency ecosystem from 2020 to 2025.

Methodology
The authors construct time-dependent, directed, and weighted networks based on high-frequency price data obtained from the Kraken Crypto Exchange. The methodology proceeds through the following steps:

Data Preprocessing:
- Source: Trade-level data (timestamps, prices in USD, volumes) from 2020 to March 2025.
- Aggregation: Prices are aggregated into 1-minute intervals using the Volume-Weighted Average Price (VWAP).
- Returns: Logarithmic returns are calculated as $R(t) = \log P(t + \Delta t) - \log P(t)$ , where $\Delta t = 1$ minute.
- Stationarity: The Augmented Dickey–Fuller (ADF) test is applied to ensure stationarity. Approximately 99.06% of the data is confirmed strictly stationary; non-stationary instances (stale pricing) are excluded.
- Windowing: The time series is divided into non-overlapping weekly windows (10,080 data points each), resulting in 275 weekly networks.
Network Construction:
- Granger Causality (GC): Interactions are inferred using the GC framework. For each pair of assets $(X, Y)$ , a restricted autoregressive model (using only $Y$ 's past) is compared against a full model (including $X$ 's past).
- Weight Definition: The GC strength is quantified as the log-variance ratio of the residuals: $G_{X \to Y} = \ln(\sigma^2_r / \sigma^2_f)$ .
- Significance Testing: To address the multiple-testing problem inherent in pairwise comparisons ( $N(N-1)$ tests), the Benjamini–Hochberg (BH) procedure is applied to control the False Discovery Rate (FDR) at $q=0.05$ . Links are retained only if their p-values satisfy the BH threshold.
- Robustness Check: To ensure links reflect genuine pairwise influence rather than co-movement driven by a common market shock, the authors repeat the analysis after regressing out the first principal component (PC1) of the return matrix.
Analysis:
- Statistical properties of returns (heavy tails) are analyzed using Complementary Cumulative Distribution Functions (CCDF).
- Network heterogeneity is assessed via link weights and nodal strengths (in-strength $s_{in}$ and out-strength $s_{out}$ ).
- Temporal evolution is tracked by ranking assets based on their average quarterly out-strength.

Key Results

Statistical Properties of Returns: Normalized 1-minute log-returns exhibit heavy-tailed distributions consistent with stylized facts of financial markets. Both positive and negative tails follow a power-law form $P(r) \sim r^{1-\gamma}$ . The estimated exponents ( $\gamma \approx 3.66 - 4.08$ ) are higher than those reported in earlier studies (e.g., $\gamma \approx 3$ for Bitcoin), suggesting structural changes and market maturation.
Network Heterogeneity: The resulting networks display pronounced heterogeneity. Both link weights and nodal strengths follow broad, heavy-tailed distributions, indicating that a small subset of cryptocurrencies contributes disproportionately to the total flow of influence.
Scaling Relationship: A strong positive correlation exists between nodal out-strength ( $s_{out}$ ) and in-strength ( $s_{in}$ ). However, the relationship is sublinear, following a power law $s_{out} \sim s_{in}^{\alpha}$ with $\alpha = 0.909$ . This indicates that while highly connected nodes are influential in both directions, the increase in outgoing influence is slower than linear relative to incoming influence. This sublinear scaling is confirmed to be a structural property, not an artifact of asset liquidity (traded volume).
Temporal Dynamics and Ranking:
- Ethereum (ETH): Consistently occupies the top position in influence rankings throughout the 2020–2025 period, reflecting sustained dominance.
- Bitcoin (XBT): Shows a gradual decline in relative importance over time.
- Volatility: The ranking structure exhibits substantial temporal variability. Unlike "super-stable" nodes found in other complex systems, the top positions are frequently occupied by different assets. Over the five-year span, 17 distinct cryptocurrencies appeared in the top five.
- Market Events: The composition of top-ranked assets shifts significantly, particularly between Q3 2022 and Q1 2025, coinciding with events such as the collapse of FTX and regulatory actions (e.g., the SEC lawsuit against Ripple).
Robustness: The network structure, including nodal strengths and link weights, remains virtually unchanged (Spearman correlation $\rho > 0.99$ ) after removing the common market factor (PC1), confirming that the inferred Granger causal links represent genuine pairwise interactions.

Significance and Claims
The paper claims to provide a more rigorous and temporally detailed reconstruction of cryptocurrency influence networks compared to prior work (specifically Scagliarini et al., 2021). By extending the analysis from a 2-year window to 5 years and increasing the asset universe from ~30 to ~390 coins, the study reveals a highly competitive and non-stationary organization of the cryptocurrency ecosystem.

Key contributions include:

Dynamic Hierarchy: Uncovering a dynamically evolving hierarchy where dominance shifts from Bitcoin to Ethereum, contrasting with findings of stable hierarchies in other systems.
Sublinear Scaling: Identifying a previously unreported nontrivial sublinear scaling relationship between in-strength and out-strength, pointing to an inherent asymmetry in influence propagation.
Methodological Rigor: Demonstrating the effectiveness of combining high-frequency data with strict stationarity testing and FDR correction to map the flow of influence.

The authors conclude that the cryptocurrency market is intrinsically dynamic, with influence continuously redistributed among assets, and that network-based frameworks are effective tools for identifying these shifting drivers of market behavior.