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

Abhijit Chakraborty; Mahesh Peyyala; Shubhangam Shukla

REVIEW 2 major objections 5 minor 2 cited by

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

T0 review · 2 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Weekly Granger-causality networks of 1-minute crypto returns show Ethereum as the persistent top influencer while Bitcoin declines and the top-five ranks turn over repeatedly.

desk verdict Solid longer-horizon GC-network extension of Scagliarini et al.; ETH-dominant ranking with high turnover is the real empirical payload, but N-growth + fixed FDR makes the time series only partly comparable. read the letter →

arxiv 2606.25466 v2 pith:CXPCZSE7 submitted 2026-06-24 q-fin.TR q-fin.GN

Shubhangam Shukla , Mahesh Peyyala , Abhijit Chakraborty This is my paper · ORCID

classification q-fin.TRq-fin.GN

keywords cryptocurrencieslog-returnGrangercausalityweightednetworkshigh-frequencydatanodalout-strengthmarketinfluencehierarchy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper maps how influence flows among cryptocurrencies by building directed, weighted networks from statistically significant Granger causal links between their one-minute log-returns, week by week from 2020 to early 2025. The authors show that the resulting networks are highly heterogeneous: a few assets account for most of the outgoing and incoming influence, and the ranking of assets by out-strength changes substantially over time. Ethereum stays at the top of that ranking throughout the five-year window, Bitcoin’s relative position erodes, and seventeen different coins appear in the top five at least once. A sympathetic reader cares because the result replaces the image of a stable, Bitcoin-centered hierarchy with a picture of a competitive, non-stationary market in which leadership continually shifts. The same analysis also recovers the familiar heavy-tailed return distributions and documents a sub-linear scaling between in-strength and out-strength that is not explained by trading volume.

What carries the argument

Time-dependent directed weighted networks whose edges are the log-variance-ratio Granger-causality strengths (after ADF stationarity screening and Benjamini–Hochberg FDR control at q = 0.05) computed on one-minute log-returns inside non-overlapping weekly windows; nodal out-strength is then used as the ranking measure of influence.

What would settle it

Recompute the same weekly networks with a non-linear or transfer-entropy causality measure (or with higher-order residualization beyond the first principal component) and check whether Ethereum’s continuous top ranking and the observed top-five turnover of seventeen assets disappear.

Watch

Extended reading notes

Core claim

Ranking cryptocurrencies by nodal out-strength in weekly Granger-causality networks of one-minute log-returns reveals a dynamically evolving hierarchy: Ethereum remains the most influential asset across 2020–2025, Bitcoin’s relative influence declines, and seventeen distinct assets occupy the top-five positions, demonstrating a competitive and non-stable organization of market influence.

Load-bearing premise

That statistically significant linear Granger causality on one-minute returns, even after stationarity filtering and false-discovery control, is a faithful measure of directed influence rather than an artifact of residual common factors or the linear VAR specification itself.

Editorial extensions

If this is right

Market-influence rankings cannot be treated as stable; any monitoring system must be recomputed on short horizons.
Ethereum’s sustained out-strength dominance supplies a quantitative counterpart to narratives of its technological maturation.
Bitcoin’s gradual loss of relative out-strength indicates that market leadership is not locked to the oldest or largest asset.
The sub-linear in-strength–out-strength scaling implies an intrinsic asymmetry in how influence is received versus transmitted across the crypto market.
Seventeen distinct top-five occupants over five years quantify the competitive turnover of the ecosystem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If out-strength rankings continue to turn over this rapidly, portfolio-risk models that treat a fixed set of “systemic” cryptos as permanent hubs will systematically mis-estimate contagion paths.
The same weekly Granger pipeline could be applied to traditional equity or FX markets to test whether the absence of super-stable nodes is crypto-specific or a general high-frequency phenomenon.
Combining the price-based influence networks with wallet-level transaction networks (as the authors themselves suggest) would allow a direct test of whether price leadership coincides with on-chain fund-flow leadership.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing worth knowing: this is a careful, longer-horizon extension of the 2020–21 Granger-causality crypto networks (Scagliarini et al.). They push the window to 2020–March 2025 (275 weekly 1-min networks), let the asset universe grow from ~30 to ~390, add ADF stationarity filtering, Benjamini–Hochberg FDR at q=0.05, and then rank by quarterly-averaged out-strength. The headline empirical result is new relative to that short-window literature: ETH stays #1, BTC declines, and 17 distinct coins cycle through the top-5. That ranking time series (Fig. 5) is the paper’s actual contribution.

What they do well is the hygiene. Stationarity is checked (99 % pass ADF), multiple testing is controlled, lag selection is BIC, and they run a PC1-removal robustness check that leaves out-strengths and link weights almost unchanged (Spearman ρ ≈ 0.999 / 0.995). The sublinear in/out-strength scaling (α ≈ 0.91, with SE and a volume covariate that does not kill it) is cleanly reported and not an artifact of liquidity. Heavy-tail exponents on the returns themselves are standard but properly estimated. Self-citation of their earlier wallet papers is light and not load-bearing.

The soft spot that actually matters is temporal comparability of the ranking. As N grows by an order of magnitude, the number of pairwise tests grows quadratically, so the BH threshold tightens while new nodes keep entering. Normalizing out-strength by contemporaneous N does not fully neutralize that. The reported BTC decline and the post-2022 turnover spike could therefore mix genuine influence shifts with changing detection power and network density. The PC1 and volume checks do not address this. Linear GC on 1-min returns is also only a statistical notion of “influence,” which the authors themselves note; that is a known limitation of the method, not a hidden flaw. No code or data release is a practical annoyance for replication, nothing more.

This is useful descriptive microstructure work for anyone who builds crypto influence or systemic-risk indicators. It does not resolve a theoretical question or change pricing practice. I would send it to peer review; a referee can ask for a fixed-core-asset robustness check and uncertainty bands on the ranks. Worth reading if you work in this niche; not a must-cite for everyone.

Referee Report

2 major / 5 minor

Summary. The manuscript constructs weekly directed, weighted networks of cryptocurrencies from statistically significant Granger-causal links among 1-minute log-returns (Kraken, Jan 2020–Mar 2025, 275 windows). After ADF stationarity filtering and Benjamini–Hochberg FDR control (q=0.05), link weights are the log residual-variance ratios of restricted versus full VARs (BIC lag selection). The authors report heavy-tailed normalized returns, heavy-tailed weight and strength distributions, a sublinear scaling sout∼sin^α (α≈0.91) that survives a volume covariate, and a quarterly ranking by N-normalized out-strength in which Ethereum remains #1, Bitcoin declines, and 17 distinct assets occupy the top five, interpreted as a competitive, non-stable hierarchy of influence. A PC1-removal robustness check leaves out-strengths and weights essentially unchanged (Spearman ρ≈0.999/0.995).

Significance. If the temporal rankings are comparable, the paper supplies a longer-horizon, higher-resolution extension of earlier GC-network studies of crypto (notably Scagliarini et al.), documenting a clear shift of dominance from Bitcoin toward Ethereum and substantial top-five turnover. Strengths include systematic stationarity testing, explicit FDR control, BIC lag selection, a volume-controlled regression for the sublinear scaling, and a transparent common-factor robustness check. These elements make the work a useful empirical contribution to the network analysis of high-frequency crypto markets, provided the comparability of the ranking measure across a growing asset universe is secured.

major comments (2)

Results §4 / Fig. 5: the central ranking claim rests on quarterly averages of out-strength normalized by contemporaneous N. Because N grows from ~30 to ~390, the number of pairwise tests rises by two orders of magnitude while the BH threshold (q=0.05) is held fixed; detection power and network density therefore change systematically. The /N normalization does not automatically guarantee that relative out-strengths of incumbents remain comparable. Without a fixed-core or density-matched robustness check, the reported Bitcoin decline and elevated post-2022 turnover may partly reflect the expanding universe and tightening multiple-testing correction rather than pure reorganization of influence.
Methods §3 and Results §4: linear pairwise Granger causality on 1-minute returns is treated as a faithful measure of directed “influence.” Although the PC1 filter addresses a shared market factor, residual non-stationarity after ADF filtering, possible nonlinear dependence, and the linear VAR specification itself remain unexamined. At least one additional check (e.g., nonlinear GC or a coarser sampling frequency) is needed to support the interpretive leap from statistical predictability to market influence that underpins the hierarchy narrative.

minor comments (5)

Fig. 1: power-law exponents are given for only two assets; a short table or statement that the remaining coins yield comparable γ would strengthen the stylized-fact claim.
Eq. (4) and Table I: the variance-reduction percentage and GX→Y are reported for a single illustrative pair; a brief distribution of typical GC strengths across weeks would help the reader gauge effect sizes.
Fig. 5 caption: the symbol list is helpful but lengthy; a supplementary table mapping tickers to full names would improve readability.
Introduction and Conclusions: several self-citations to the authors’ wallet-network papers are appropriate but could be condensed to avoid redundancy with the price-based focus of the present work.
Typographical inconsistencies appear in author names and affiliations (e.g., “P eyy ala”, “C hakraborty”, “Insti tutes”); these should be corrected in production.

Circularity Check

0 steps flagged · score 1.0 of 10

Empirical GC-network construction and out-strength ranking; no derivation reduces to its inputs by construction. Minor non-load-bearing self-citations of authors' prior wallet-network papers.

full rationale

The paper's central claims are empirical summaries of directed weighted networks built from pairwise linear Granger causality (Eqs. 2-4) on 1-minute log-returns, after ADF stationarity filtering and Benjamini-Hochberg FDR control (Eq. 5). Nodal out-strength is the sum of retained GC weights; quarterly ranking (Fig. 5) is simply the time-averaged, N-normalized ordering of those strengths. Nothing is predicted from a fitted parameter that is then re-used as input, nor is any uniqueness or scaling relation imported by self-citation as a forced premise. The sublinear sout ~ sin^alpha fit (Fig. 4, alpha=0.909) and the PC1-residual robustness check (Fig. 6) are post-hoc observations, not circular predictions. Self-citations [5-9] appear only in the introduction as complementary wallet-level work and do not enter the price-based GC pipeline or the ranking result. The comparison to Scagliarini et al. [4] is external. Methodological concerns about changing N and FDR power affect temporal comparability but are not circularity. Derivation chain is self-contained against the data.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central ranking claim rests on standard econometric assumptions (linear VAR, Granger definition, FDR control) plus a few free choices (weekly windows, max lag 10, FDR q=0.05, out-strength as the influence proxy). No new physical entities are postulated; the only free parameters are the conventional statistical thresholds and the lag-selection bound.

free parameters (4)

FDR level q = 0.05
Benjamini–Hochberg threshold set to 0.05; controls which GC links enter the network and therefore which out-strengths are computed.
maximum lag for BIC = 10
Upper bound p≤10 used when selecting the VAR lag order; affects residual variances and GC strengths.
weekly window length = 10080 minutes
Non-overlapping 10 080-minute windows define the temporal resolution of the network sequence and the quarterly averages.
ADF significance level = 0.01
α=0.01 used to declare stationarity; 0.94 % of series excluded.

assumptions (4)

domain assumption Linear Granger causality on log-returns quantifies directed predictive influence between assets.
Invoked throughout §3 and used as the sole link-construction rule; the paper notes it is statistical, not mechanistic.
domain assumption Log-returns that pass the ADF test at α=0.01 are sufficiently stationary for VAR estimation.
Stated in §2; 99.06 % of series retained on this basis.
standard math Benjamini–Hochberg FDR at q=0.05 adequately controls false-positive directed links under the multiple-testing regime.
Applied independently each week (§3) to the N(N−1) pairwise p-values.
ad hoc to paper Nodal out-strength is a valid scalar proxy for a cryptocurrency’s overall market influence.
Chosen as the ranking metric in §4 without comparison to eigenvector centrality, PageRank, or other network measures.

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Cite this review

Pith. "Pith review of Time-dependent weighted directed networks of cryptocurrency interaction from high-frequency returns." pith.science (2026). https://pith.science/paper/CXPCZSE7

@misc{pith2026260625466,
  author       = {Pith},
  title        = {Pith review of: Time-dependent weighted directed networks of cryptocurrency interaction from high-frequency returns},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CXPCZSE7}},
  note         = {Machine review of arXiv:2606.25466}
}

read the original 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.

Figures

Figures reproduced from arXiv: 2606.25466 by the authors.

**Figure 1.** Complementary cumulative distribution functions (CCDF) of the normalized 1-minute log-returns r. Panels (a) and (b) show the positive and negative tails for XBT, respectively, while panels (c) and (d) correspond to XRP. The red lines represent power-law fits of the form P(r) ∼ r 1−γ , with estimated exponents γ = 3.91, 3.78, 4.08, and 3.66 for panels (a)–(d), respectively. The exponents are obtained using the maxim… view at source ↗

**Figure 2.** (a) Normalized log-return of ETH (grey), along with the restricted model (red) defined in Eq. 2, where the target variable Y represents log-return time series for ETH, and the full model (blue) defined in Eq. 3, where X represents log-return time series for XBT. In both models, the error term is excluded. Panels (b) and (c) show the corresponding residuals (error terms) for the full model and restricted model, respe… view at source ↗

**Figure 3.** Complementary cumulative distribution functions (CCDF) of key network measures: (a) link weight w, (b) nodal out-strength sout, and (c) nodal in-strength sin. Results are shown for a representative week (24–30 March 2025); other weeks exhibit qualitatively similar behaviour. presents the distribution of link weights w, while panels (b) and (c) show the distributions of nodal out-strength sout and in-strength sin, re… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Scatter plot of nodal out-strength sout versus in-strength sin for all nodes. The red line represents a power-law fit of the form sout ∼ s α in . The estimated exponent α = 0.91 indicates a nontrivial sublinear relationship between the two quantities. Results are shown…

**Figure 5.** Figure 5: Ranking of cryptocurrencies based on their average quarterly out-strength (influence) over the period 2020–2025. Ethereum (ETH) consistently ranks as the most influential cryptocurrency, while Bitcoin (XBT) shows a gradual decline in influence. The rankings exhibit sig…

**Figure 6.** Figure 6: Robustness of the inferred network structure after removing the common market factor via first principal component (PC1) regression. Panel (a) compares the nodal out-strength sout computed from the original log-returns (PC unregressed) against those computed from the…

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Pith tools

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