pith. sign in

arxiv: 2606.20566 · v1 · pith:S7FGVOREnew · submitted 2026-04-21 · 💻 cs.NI · cs.SI

Shared Channel Capacity and Node Lifetime: An Empirical Study of the Lightning Network

Danila Valko , Jorge Marx G\'omez This is my paper

Pith reviewed 2026-07-05 08:38 UTC · model glm-5.2

classification 💻 cs.NI cs.SI
keywords nodecapacitylifetimechannelnetworkdegreeeconomicliquidity
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The pith

Connectivity, not time, drives Lightning Network liquidity

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

This paper argues that in the Lightning Network, a node's shared channel capacity is driven primarily by how many connections it has (its degree), not by how long it has been alive. Using five years of topology snapshots (2019-2023), the authors show that while longer-lived nodes do hold more capacity, this relationship is almost entirely mediated by node degree: longer lifetime leads to more connections, and more connections lead to greater capacity. When degree is statistically controlled for, the direct effect of lifetime on capacity actually reverses sign and becomes negative. The paper also finds that country-level economic conditions (proxied by GDP per capita) significantly influence capacity, and that hierarchical mixed-level models accounting for geographic nesting outperform flat regressions. The moderation analysis further shows that the lifetime-capacity relationship is heterogeneous: for highly connected, high-capacity nodes, longer lifetime amplifies capacity accumulation, while for low-degree nodes the direct effect is weak or negative.

Core claim

The central finding is a mediation effect: node degree transmits the positive relationship between node lifetime and shared channel capacity. Longer-lived nodes accumulate more connections (β≈0.09-0.13, p<0.001), and higher degree strongly predicts capacity (β≈1.9-2.3, p<0.001). Once degree is included as a control, the lifetime coefficient reverses sign (β≈-0.08 to -0.15, p<0.001), meaning that holding connectivity constant, time itself is associated with lower capacity. The total positive effect of lifetime on capacity thus runs through the connectivity channel, not through direct temporal accumulation.

What carries the argument

The analytical machinery is a mediation-moderation regression framework applied to log-transformed node-level variables. The mediation claim is established through a two-step procedure: first regressing degree on lifetime, then regressing capacity on both degree and lifetime and observing the sign reversal of the lifetime coefficient. Robustness is tested via OLS, robust linear models (M-estimators), quantile regression, country/region fixed effects, mixed-level hierarchical models with random intercepts, and an explicit GDP-per-capita control.

If this is right

  • Node operators seeking to maximize routing capacity should prioritize expanding their number of channel connections rather than simply staying online longer, since the mediation analysis shows degree is the operative variable for liquidity accumulation.
  • Protocol-level incentive mechanisms that reward connectivity formation (especially for new or geographically underrepresented nodes) may be more effective at redistributing liquidity than measures aimed at prolonging node lifetime.
  • The finding that GDP per capita correlates with node capacity but not with degree suggests that economic conditions affect how much liquidity a node can commit to channels but not how many connections it forms, pointing to capital access as a binding constraint distinct from network topology.
  • The sign reversal of the lifetime coefficient when degree is controlled implies that among nodes with identical connectivity, older nodes may actually hold less capacity, possibly due to channel aging, outdated fee structures, or failure to rebalance.

Load-bearing premise

The mediation analysis assumes that the causal chain runs from lifetime to degree to capacity, but all three variables are constructed as cross-sectional averages (degree and capacity averaged across snapshots, lifetime as the span between first and last appearance). This design collapses the temporal ordering needed to support a causal mediation claim, so the finding could equally reflect reverse causation, where well-connected, high-capacity nodes simply tend to persist for

What would settle it

If a longitudinal (panel) analysis tracking the same nodes over time showed that degree and capacity do not increase with node age after controlling for initial conditions, or if it showed that high-degree and high-capacity nodes are formed early and then persist longer (rather than degree growing over time), the mediation claim would be undermined.

Figures

Figures reproduced from arXiv: 2606.20566 by Danila Valko, Jorge Marx G\'omez.

Figure 1
Figure 1. Figure 1: Conceptual Model Preprint version – Not for redistribution [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Node Lifetime, Degree and Capacity Distribution [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Node Lifetime, Degree and Capacity Correlation [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 𝛽3 coefficients for Lifetime × Degree term using QLM, RLM, and OLS. Note. All variables are log-transformed. The shaded area beneath the fitted curves represents the 95% confidence intervals. scale-free properties and preferential attachment dynamics (e.g., [20, 22]). The strong association between degree and capacity also aligns with prior evidence of liquidity concentration among highly connected hub nod… view at source ↗
read the original abstract

The Lightning Network (LN) is a rapidly evolving payment channel network that enables scalable, off-chain transactions on top of Bitcoin. While prior research has documented its topological structure and liquidity concentration, the joint relationships between node lifetime, connectivity, and capacity remain insufficiently understood. This study provides a comprehensive empirical analysis of these relationships using a large-scale dataset of LN topology snapshots spanning the period 2019-2023. We examine whether node lifetime influences shared channel capacity, and whether this effect is mediated and moderated by node degree. In addition, we account for hierarchical geographic structure and explore the role of country-level economic conditions. The results show that node lifetime has a positive but relatively modest direct effect on capacity. This relationship is largely mediated by node degree, indicating that liquidity accumulation primarily occurs through increased connectivity. Furthermore, the interaction between lifetime and degree reveals significant heterogeneity, with stronger effects observed among highly connected and high-capacity nodes. Mixed-level models demonstrate superior explanatory power, highlighting the importance of country- and region-level variation. The inclusion of GDP per capita confirms that broader economic conditions significantly influence capacity distribution. Overall, the findings suggest that liquidity in the LN emerges from the interplay of temporal dynamics, network structure, and economic context. This study contributes to a more integrated understanding of payment channel networks and provides a foundation for future research on their evolution and efficiency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 7 minor

Summary. This manuscript empirically examines the relationships between node lifetime, degree (connectivity), and shared channel capacity in the Lightning Network (LN) using a large-scale dataset of topology snapshots spanning 2019–2023. The authors specify and estimate a series of regression models (OLS, robust, quantile, and mixed-level) to test three hypotheses: (H1) lifetime positively affects capacity, (H2) degree mediates the lifetime-capacity relationship, and (H3) degree moderates the lifetime-capacity relationship. The study finds that while lifetime has a modest positive total effect on capacity, this effect is largely mediated by degree, with the direct effect of lifetime reversing sign upon inclusion of degree. The paper also incorporates country-level fixed/random effects and GDP per capita, finding that geographic and economic context significantly influence capacity distribution. The dataset and geolocation pipeline are drawn from the authors' prior work [26, 27, 28] and have been validated against external benchmarks.

Significance. The paper tackles a relevant gap in the LN literature by moving beyond descriptive topology studies to a multivariate explanatory framework integrating temporal, structural, and geographic factors. The use of mixed-level models and the inclusion of GDP per capita as a country-level control are methodological strengths that add value over prior pairwise association studies. The authors provide a public repository for code and data, enhancing reproducibility. The robustness strategy—comparing OLS, RLM, QLM, and MLM specifications—is appropriate and thorough. However, the central mediation claim is subject to a structural concern regarding the definitional relationship between the mediator and the dependent variable, which must be addressed for the findings to be fully convincing.

major comments (3)
  1. §4.2.1, Eq. (1) and §5.3.2, Table 6: The mediation analysis (H2) is affected by a mechanical definitional link between the mediator (degree) and the outcome (capacity). Capacity is defined in Eq. (1) as (1/2)·Σ capacity[e] over incident edges, while degree is the count of those edges. In log space, log(capacity) ≈ log(degree) + log(avg_channel_capacity). This means that including degree in the H2'' regression (Table 6) will absorb variation in capacity by construction, making the sign reversal of the lifetime coefficient a near-mathematical necessity rather than evidence of a substantive mediation mechanism. The coefficient on degree (β≈1.9–2.3) exceeding 1.0 indicates super-linear scaling beyond the mechanical baseline, which is interesting, but the paper does not discuss or account for the mechanical relationship. The authors should explicitly acknowledge this definitional link and re-
  2. §4.2.1 and §5.3.2: The mediation analysis assumes temporal ordering (lifetime → degree → capacity), but all three variables are constructed as cross-sectional node-level summaries: lifetime = last_snapshot − first_snapshot, while degree and capacity are per-node averages across all snapshots. This construction collapses within-node temporal dynamics. A node that established high degree and high capacity early could persist longer precisely because it was well-connected (reverse causation: degree → lifetime), which would produce the same coefficient pattern observed in Tables 5–6 without any mediation occurring. The paper acknowledges in §6.2 that 'the empirical framework is based on statistical associations rather than causal identification,' but the mediation framing in H2 and the causal language throughout ('transmission channel,' 'accumulation mechanism') exceed what the cross-
  3. §5.3.3, Table 7: The moderation analysis (H3) reports opposite signs for the interaction term across subsamples—negative for non-geolocated nodes (β≈−0.035 to −0.046) and positive for geolocated nodes (β≈0.04–0.08). The paper attributes this to possible Simpson's paradox or 'systematically different connectivity dynamics,' but does not test this claim. Given that the non-geolocated subsample is 3.5× larger, the practical implication of this sign reversal for the overall network is unclear. The authors should either provide a formal test for why the sign flips or substantially temper the claim that H3 is supported.
minor comments (7)
  1. §5.3.1: The text states 'β≃ 0.05—2.1' for the lifetime coefficient, but Table 4 shows values ranging from 0.045 to 0.212. The '2.1' appears to be a typo for '0.21'.
  2. §4.2.2: The filtering removes nodes with zero degree (1.8%), zero capacity (6.6%), or zero lifetime (1.8%), but it is unclear whether these categories overlap. Reporting the joint distribution or the total percentage removed would improve clarity.
  3. Table 4: The R² values for H1 models range from 0.001 to 0.11, which the authors honestly acknowledge as 'limited.' However, the abstract's claim that 'node lifetime has a positive but relatively modest direct effect' could be strengthened by noting the near-zero explanatory power in the abstract itself.
  4. §5.3.4, Table 8: The sample size drops from 7,917 (Tables 4–7) to 7,893 when GDP per capita is included. The reason for this drop (likely missing GDP data for some countries) should be noted.
  5. Fig. 2: The x-axis label for the capacity histogram reads 'Capacity [sat] ×10^9' but the tick marks range from 0 to 4, suggesting the axis is in billions of satoshis. This should be clarified, as raw satoshi values would be much larger.
  6. §4.3.1: The paper uses 'mediation' and 'moderation' in the standard Baron and Kenny sense but does not cite the foundational mediation/moderation literature (e.g., Baron & Kenny 1986; Hayes 2013). Including at least one reference would situate the analytical approach.
  7. §6.1: The practical implications section suggests node operators should 'prioritize... the timely expansion of connectivity,' but given the cross-sectional, non-causal nature of the data, this recommendation should be stated more cautiously.

Simulated Author's Rebuttal

3 responses · 2 unresolved

We thank the referee for a careful and constructive review. The referee raises three substantive points: (1) a mechanical/definitional link between degree and capacity that may inflate the mediation result, (2) temporal ordering concerns given the cross-sectional construction of variables, and (3) unexplained sign reversal of the moderation interaction across subsamples. We agree that all three points identify genuine limitations of the current manuscript. We will revise the manuscript to explicitly acknowledge the mechanical relationship, substantially temper the causal and mediation language, and either provide a formal test for the sign reversal or downgrade the H3 claim. Details are below.

read point-by-point responses

  1. Referee: §4.2.1, Eq. (1) and §5.3.2, Table 6: The mediation analysis (H2) is affected by a mechanical definitional link between the mediator (degree) and the outcome (capacity). Capacity is defined as (1/2)·Σ capacity[e] over incident edges, while degree is the count of those edges. In log space, log(capacity) ≈ log(degree) + log(avg_channel_capacity). Including degree in the H2'' regression will absorb variation in capacity by construction, making the sign reversal of the lifetime coefficient a near-mathematical necessity rather than evidence of a substantive mediation mechanism. The coefficient on degree (β≈1.9–2.3) exceeding 1.0 indicates super-linear scaling beyond the mechanical baseline, which is interesting, but the paper does not discuss or account for the mechanical relationship.

    Authors: The referee is correct, and we acknowledge this as a genuine limitation of the current manuscript. As the referee notes, because capacity is defined as the sum of per-edge capacities divided by two, and degree is the count of those edges, log(capacity) = log(degree) + log(average per-channel capacity). This means that including degree as a regressor for log(capacity) will mechanically absorb a large portion of the variation by construction. We agree that the sign reversal of the lifetime coefficient upon inclusion of degree is therefore not, by itself, convincing evidence of a substantive mediation mechanism—it is largely a mathematical consequence of the definitional link. We will revise the manuscript to explicitly acknowledge this mechanical relationship in both the methods section and the discussion of Table 6. We will also reframe the interpretation: the coefficient on degree exceeding 1.0 (β≈1.9–2.3) indicates super-linear scaling of capacity with connectivity beyond the mechanical baseline, which is a substantively interesting finding about preferential attachment and hub concentration, but it does not constitute clean evidence of mediation in the causal sense. We will add a decomposition showing how much of the degree–capacity association is mechanical versus substantive, and we will substantially temper the mediation language throughout the paper, including in the abstract, H2 framing, and discussion. We will also consider an alternative specification using average per-channel capacity as the dependent variable to remove the mechanical link, though we note this changes the research question. At minimum, the revised manuscript will explicitly state that the mediation analysis is confounded by the definitional relationship and that the H2 results should be интерп revision: yes

  2. Referee: §4.2.1 and §5.3.2: The mediation analysis assumes temporal ordering (lifetime → degree → capacity), but all three variables are constructed as cross-sectional node-level summaries: lifetime = last_snapshot − first_snapshot, while degree and capacity are per-node averages across all snapshots. This construction collapses within-node temporal dynamics. A node that established high degree and high capacity early could persist longer precisely because it was well-connected (reverse causation: degree → lifetime), which would produce the same coefficient pattern observed in Tables 5–6 without any mediation occurring. The paper acknowledges in §6.2 that 'the empirical framework is based on statistical associations rather than causal identification,' but the mediation framing in H2 and the causal language throughout ('transmission channel,' 'accumulation mechanism') exceed what the cross-

    Authors: The referee is correct that the cross-sectional construction of all three variables collapses within-node temporal dynamics and does not establish the temporal ordering required for a causal mediation claim. Lifetime is defined as last_snapshot minus first_snapshot, while degree and capacity are per-node averages across all snapshots. This means the analysis cannot distinguish the proposed pathway (lifetime → degree → capacity) from the reverse causal pathway the referee identifies (degree → lifetime, where well-connected nodes persist longer precisely because they are well-connected). We agree that the causal language used throughout the manuscript—terms such as 'transmission channel,' 'accumulation mechanism,' and 'mediates'—exceeds what the cross-sectional framework can support. In the revised manuscript, we will (1) reframe H2 as an exploratory test of statistical association rather than causal mediation, (2) replace causal language with associational language throughout (e.g., 'degree accounts for the association between lifetime and capacity' rather than 'degree transmits the effect of lifetime to capacity'), (3) explicitly discuss the reverse causation concern in the limitations section, and (4) note that a proper causal mediation analysis would require panel data with time-lagged variables, which we leave for future work. We will also add a note that the cross-sectional averages may mask important within-node temporal dynamics that a panel or survival analysis framework could better capture. We cannot fully resolve this concern within the current data and design, and we will be transparent about that. revision: yes

  3. Referee: §5.3.3, Table 7: The moderation analysis (H3) reports opposite signs for the interaction term across subsamples—negative for non-geolocated nodes (β≈−0.035 to −0.046) and positive for geolocated nodes (β≈0.04–0.08). The paper attributes this to possible Simpson's paradox or 'systematically different connectivity dynamics,' but does not test this claim. Given that the non-geolocated subsample is 3.5× larger, the practical implication of this sign reversal for the overall network is unclear. The authors should either provide a formal test for why the sign flips or substantially temper the claim that H3 is supported.

    Authors: The referee is correct that the sign reversal of the interaction term across subsamples is currently under-investigated. We offer two responses. First, we note that the quantile regression analysis in §5.3.5 (Figure 4) already provides partial evidence that the apparent inconsistency reflects genuine distributional heterogeneity rather than a pure sampling artifact: across both subsamples, the interaction coefficient trends upward across quantiles, becoming less negative (and in some cases positive) at higher capacity deciles. This suggests the subsample-level sign difference is partly driven by differences in the capacity distribution composition between geolocated and non-geolocated nodes. However, we agree that this is not a formal test of the sign flip. In the revised manuscript, we will either (a) conduct a formal test—for example, by estimating a pooled model with a subsample indicator and its three-way interaction with lifetime and degree, or by applying a Chow-type test for coefficient equality across subsamples—or (b) if such a test does not conclusively explain the reversal, substantially temper the claim that H3 is supported. We are inclined toward option (a) as a first step, but if the test is inconclusive, we will revise the H3 conclusion to state that the moderation effect is present but directionally unstable across subsamples, and that the evidence for H3 is therefore mixed rather than supportive. We agree that the current language ('provide support for H3') is too strong given the sign reversal, and we will revise accordingly. revision: partial

standing simulated objections not resolved
  • The mechanical/definitional link between degree and capacity (Comment 1) cannot be fully resolved without either changing the dependent variable (e.g., to average per-channel capacity) or adopting a fundamentally different analytical framework. We can acknowledge and partially address it, but the core structural issue is inherent to the variable definitions.
  • The temporal ordering concern (Comment 2) cannot be resolved within the current cross-sectional data construction. A proper causal mediation analysis would require panel data with time-lged variables, which is beyond the scope of the current manuscript.

Circularity Check

0 steps flagged

No significant circularity; self-cited dataset is externally validated and no prediction reduces to its inputs by construction

full rationale

The paper's central claims are derived from standard regression, mediation, and moderation analyses applied to an observational dataset. The dataset (refs [26, 27]) is authored by the same authors but is validated against external benchmarks (BitcoinVisuals, mempool.space) with stated accuracy metrics in §4.1, and the GDP per capita data comes from the World Bank [31]. No fitted parameters are renamed as predictions, and no derivation step reduces to its inputs by definition. The mechanical relationship between degree (edge count) and capacity (sum over the same edges, Eq. 1) creates a baseline correlation that inflates the degree-capacity coefficient, but the estimated coefficient (β≈1.9–2.3) exceeds the mechanical benchmark of 1.0, indicating the result contains information beyond the definitional link. This is a methodological limitation (correctness risk), not circularity. The self-citations are for data provision and a literature review, not for a load-bearing theoretical premise or uniqueness theorem. Score 1 reflects the minor self-citation for the dataset, which is not load-bearing for the central claims.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The paper introduces no new entities, particles, forces, or theoretical constructs. It is a purely empirical study using standard statistical methods (OLS, RLM, QLM, mixed-level models) applied to observational data. The free parameters are standard regression coefficients. The axioms are domain assumptions about data representativeness and variable construction, with one ad hoc assumption about the suitability of cross-sectional averages for mediation analysis.

free parameters (1)
  • None (fitted regression coefficients) = Various β coefficients reported in Tables 4–8
    All coefficients are standard regression estimates (OLS, RLM, QLM, MLM) fitted to the data. No ad hoc parameters are introduced to make a derivation work. The models are standard statistical specifications without free parameters beyond the estimated coefficients.
axioms (4)
  • domain assumption Gossip messages and reconstructed snapshots accurately represent the LN topology at each time point.
    §4.1: The dataset is reconstructed from 35M gossip messages. The authors validate against external benchmarks but acknowledge inherent limitations of observational data.
  • domain assumption Geolocated subsample (declining from ~40% to ~15% over time) is representative of the full network.
    §4.1 and §5.1: K-S statistics (0.13–0.19) are cited as evidence of representativeness, but the declining coverage rate and TOR usage create potential selection bias.
  • ad hoc to paper Cross-sectional averages of degree and capacity across snapshots can be used in mediation analysis that assumes temporal ordering (lifetime → degree → capacity).
    §4.2.1: Degree and capacity are averaged across all snapshots per node, collapsing temporal dynamics. The mediation framework in H2 requires temporal precedence, which the averaged variables do not provide.
  • domain assumption GDP per capita is a valid proxy for country-level economic, infrastructural, and environmental conditions relevant to LN capacity.
    §4.3.3: The authors justify this by citing correlations with HDI, GCI, and EPI, but acknowledge other factors (regulatory frameworks, energy costs) are not included.

pith-pipeline@v1.1.0-glm · 24283 in / 2990 out tokens · 219563 ms · 2026-07-05T08:38:42.446658+00:00 · methodology

discussion (0)

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Reference graph

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