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

Abstract: We investigate the evolving structure of interactions in cryptocurrency markets using a network-based framework constructed from high-frequency price data spanning 2020-2025. Directed and weighted networks are constructed from statistically significant Granger causal relationships between cryptocurrency log-returns, enabling us to quantify the flow of influence across assets. We find that normalized returns exhibit heavy-tailed distributions, consistent with the presence of large intermittent fluctuations and in line with stylized facts of financial markets. The resulting networks display pronounced heterogeneity in link weights and nodal strengths, indicating that a small subset of cryptocurrencies contributes disproportionately to market dynamics. By ranking cryptocurrencies based on their nodal out-strength, we uncover a dynamically evolving hierarchy of influence. Ethereum consistently emerges as the most influential asset, while Bitcoin shows a gradual decline in its relative importance. The ranking structure exhibits substantial temporal variability, with multiple cryptocurrencies entering and exiting the top positions over time. Our findings reveal a highly competitive and non-stable organization of the cryptocurrency ecosystem.

• The paper constructs weekly Granger-causal cryptocurrency networks from one-minute Kraken returns, using stationarity screening and Benjamini–Hochberg false-discovery control across 275 windows.
• The paper finds heterogeneous influence structures, sublinear in-strength–out-strength scaling with α = 0.909, and a hierarchy robust to removal of the common market factor.
• The paper shows that influence changes substantially over time: Ethereum remains dominant, Bitcoin declines relatively, and 17 cryptocurrencies reach the top five between 2020 and 2025.

Overview and motivation

This paper constructs and analyzes time-dependent, directed, weighted networks of cryptocurrency interactions inferred from Granger causality (GC) between high-frequency log-returns, covering Kraken trade-level data from January 2020 to March 2025. Prices are aggregated to one-minute resolution via the volume-weighted average price (VWAP), yielding 275 nonoverlapping weekly windows of 10,080 points each, with the number of actively traded assets growing from roughly 30 to about 390 over the study period. The central contributions are (i) a statistically rigorous reconstruction of weekly influence networks with explicit stationarity screening and multiple-hypothesis correction, (ii) a quarterly-resolved hierarchy of asset influence via nodal out-strength, and (iii) a robustness analysis demonstrating that inferred links are not artifacts of a shared market mode.

Statistical properties of returns

Normalized one-minute log-returns display heavy-tailed complementary cumulative distributions, with power-law tails $P(r) \sim r^{1-\gamma}$. Maximum-likelihood estimates give $\gamma = 3.91$ and $3.78$ for the positive and negative tails of Bitcoin (XBT), and $\gamma = 4.08$ and $3.66$ for XRP. Notably, these exponents exceed the value of approximately $3$ reported in earlier work on Bitcoin fluctuations, which the authors attribute to structural maturation of the cryptocurrency market. The values remain broadly consistent with the heavy-tailed stylized facts documented for equities and foreign-exchange markets, supporting cross-asset universality of return distributions.

Network construction and methodology

For each ordered pair of stationary return series, a restricted autoregressive model (target regressed on its own past) is compared against a full model that includes the source's past values. The optimal lag order $p$ is selected by BIC with maximum lag 10, and the GC strength is the log-ratio of residual variances, $G_{X \to Y} = \ln(\sigma_r^2 / \sigma_f^2)$. As a concrete example, including XBT in a full model for ETH during a January 2020 week reduces residual variance by $3.24\%$, yielding $G_{\mathrm{XBT} \to \mathrm{ETH}} = 0.033$ with an F-statistic of $\gamma = 3.91$0 ($\gamma = 3.91$1).

Two methodological safeguards distinguish this analysis from earlier work. First, all return series are screened for stationarity using the Augmented Dickey–Fuller test at $\gamma = 3.91$2; $\gamma = 3.91$3 of series pass, and the remaining $\gamma = 3.91$4—flat, zero-variance series during extreme inactivity—are excluded. Second, because $\gamma = 3.91$5 simultaneous pairwise tests inflate the false-positive rate, the Benjamini–Hochberg false discovery rate procedure is applied within each weekly window at $\gamma = 3.91$6. The authors are explicit that GC here quantifies improved linear predictability, not mechanistic causality.

Network structure and scaling

The resulting networks are strongly heterogeneous: link weights, out-strengths, and in-strengths all exhibit broad, heavy-tailed distributions, indicating that a small subset of assets carries a disproportionate share of influence in both directions. A central quantitative finding is a sublinear power-law relationship between nodal in-strength and out-strength, $\gamma = 3.91$7 with $\gamma = 3.91$8 (SE $\gamma = 3.91$9; $3.78$0). A one-sample $3.78$1-test rejects linearity ($3.78$2, $3.78$3), with the 95% confidence interval $3.78$4 excluding unity. The authors further control for a plausible confound—highly liquid assets might simultaneously transmit and receive more influence—by adding log traded volume as a covariate; the volume coefficient is statistically insignificant ($3.78$5, $3.78$6) and $3.78$7 is essentially unchanged at $3.78$8. The sublinear scaling therefore appears to be a genuine structural asymmetry: assets with the highest incoming influence do not transmit influence proportionally.

Temporal hierarchy of influence

Ranking assets by quarterly-averaged, size-normalized out-strength reveals a markedly non-stationary hierarchy. Ethereum (ETH) holds the top position throughout 2020–2025, while Bitcoin (XBT) declines gradually in relative influence. XRP, ranked third early in the period, is displaced from 2021 onward by Cardano, Solana, and other newer platforms; the authors link this decline plausibly to the December 2020 SEC action against Ripple Labs. Solana rises sharply into the top five by 2023–2024, while Bitcoin Cash, EOS, Litecoin, and Algorand appear briefly in the early quarters and largely vanish thereafter. Ranking instability intensifies after Q3 2022, which the authors associate with the November 2022 collapse of FTX.

The most striking aggregate claim is that 17 distinct cryptocurrencies occupy a top-five position at some point over five years, with frequent entry and exit. This directly contradicts the super-stable node persistence reported for other real-world networks and, importantly, diverges from the comparatively stable hierarchy found by Scagliarini et al. over the shorter 2020–2021 window—suggesting that apparent stability in earlier studies may have been an artifact of the limited observation period.

Robustness to common market factors

A potential concern is that pairwise GC links reflect a shared market-wide shock rather than asset-to-asset interaction. To address this, the full network construction is repeated on idiosyncratic residuals obtained by regressing each asset's returns on the first principal component (market mode) of the return covariance matrix. The resulting out-strengths and link weights are nearly unchanged: Spearman rank correlations with the original quantities are $3.78$9 and $\gamma = 4.08$0, respectively, with deviations confined to weak links. The inferred influence structure is therefore robust to common-factor co-movement, and the reported hierarchy reflects genuine pairwise predictive relationships.

Limitations and open questions

Several caveats are acknowledged or implicit. Granger causality captures only linear, lagged predictability; nonlinear or contemporaneous dependence structures are not modeled. The GC framework is applied to raw returns rather than volatility or order-flow measures, so "influence" here is specific to return predictability. Explanatory narratives for ranking changes—Ethereum's proof-of-stake transition, the Ripple lawsuit, the FTX collapse—are offered as plausible associations rather than established causal mechanisms. The sublinear scaling result is demonstrated for representative weeks with qualitative similarity claimed for others, but a full longitudinal characterization of the exponent's stability is not provided. Open questions include whether higher-order dependencies (e.g., O-information) or nonlinear causality measures alter the hierarchy, and how price-based influence networks relate to wallet-level transaction networks.

Conclusion

This study delivers a statistically careful, five-year reconstruction of Granger-causal influence networks in cryptocurrency markets, combining stationarity screening, FDR correction, and principal-component robustness checks. Its principal findings are heavy-tailed return and network-strength distributions, a genuine sublinear in-strength–out-strength scaling ($\gamma = 4.08$1), and a highly competitive influence hierarchy with 17 distinct assets entering the top five—Ethereum dominant throughout, Bitcoin declining—contrasting with the super-stable structures seen in other complex systems. The results establish that the cryptocurrency influence hierarchy is intrinsically non-stationary at multi-year horizons, a conclusion that shorter-window analyses could not reach.