# A Tale of Two Layers: The Mutual Relationship between Bitcoin and Lightning Network

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Lightning Network

#### 2.2. Market Efficiency

- $0<\alpha <0.5$: long-term memory and anti-correlation;
- $0.5<\alpha <1$: long-term memory and correlation;
- $\alpha =0.5$: uncorrelated signal (no memory);
- $\alpha >1$: non-stationary signal.

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Testing for Granger-causality. The table reports the results of the Toda–Yamamoto test (Toda and Yamamoto 1995). In Panel A we test whether column variables Granger-cause the row variable, while the opposite for Panel B. To compute the DFA, we consider time windows of length $n=300$ for the first two columns and $n=600$ for the last two.

Panel A: Column G-Causes Row | BTC Alpha${}_{300}$ | BTC Vol Alpha${}_{300}$ | BTC Alpha${}_{600}$ | BTC Vol Alpha${}_{600}$ | ||

LN efficiency | statistics | 1.90 | 1.90 | 2.20 | 1.10 | |

p-value | 0.60 | 0.87 | 0.81 | 0.98 | ||

Panel B: Row G-Causes Column | BTC Alpha${}_{300}$ | BTC Vol Alpha${}_{300}$ | BTC Alpha${}_{600}$ | BTC Vol Alpha${}_{600}$ | ||

LN efficiency | statistics | 0.90 | 3.40 | 3.10 | 5.30 | |

p-value | 0.83 | 0.64 | 0.68 | 0.50 |

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**Figure 1.**Representation of different options for a multi-hop transaction. Circles represent users’ nodes while the bi-directional channels are represented with arrows in both directions.

**Figure 2.**Visual representation of the LN. The plot on the left refers to 2018/02/12, while the one on the right to 2020/08/12.

**Figure 3.**Evolution of LN’s normalized efficiency. The plot exhibits the evolution of LN’s global normalized efficiency (in light blue), its density (in black) multiplied by 10, and the median capacity installed on its channels (in red). Both density and efficiency can assume values between 0 and 1. The y-axis on the left is related to the density and efficiency measures, while the one on the right is related to the median capacity expressed in USD.

**Figure 4.**DFA exponent time evolution. The plot exhibits the DFA exponent (Peng et al. 1994, 1995) for returns (in green) and volatility (in blue). Estimates consider sliding windows of 250 daily observations and one datapoint step forward. The shadow area refers to the standard error of the corresponding coefficient. The log of Bitcoin prices (divided by ${10}^{3}$) is reported in gray. The dotted red line stands for the 0.5 level of the DFA $\alpha $ exponent.

**Table 1.**A collection of topological measures for LN. This table presents some topological measures extrapolated from the network at its first and last observation in our sample period.

12 February 2018 | 12 August 2020 | |
---|---|---|

Nodes | 538 | 7916 |

Channels | 1985 | $43,654$ |

Density | $0.014$ | $0.001$ |

Median Degree | 2 | 3 |

Average Degree | $7.37$ | $11.03$ |

Median Strength(USD) | $22.80$ | $91.70$ |

Average Strength (USD) | $211.34$ | $3523.49$ |

Average Capacity (USD) | $28.64$ | $319.47$ |

Median Capacity (USD) | $7.80$ | $57.33$ |

Total Capacity (USD) | $56,861$ | $13,945,976$ |

Assortativity | $-0.370$ | $-0.231$ |

Assortativity (W) | $-0.170$ | $-0.057$ |

Diameter | 6 | 12 |

Radius (LCC) | 4 | 6 |

Transitivity (W) | $0.120$ | $0.063$ |

Global Efficiency Norm | $0.140$ | $0.014$ |

**Table 2.**Bitcoin market ffficiency conditions. Table reports p-values for the following tests: the Runs Test (Wald and Wolfowitz 1940), the Bartels Test (Bartels 1982), the BDS Test (Broock et al. 1996), the Automatic Portmanteau Test (Escanciano and Lobato 2009), the AVR Test (Choi 1999; Kim 2009; Lo and MacKinlay 1988), and the DL Test (Domínguez and Lobato 2003). For BDS the table reports the average p-values across specifications with embedding dimensions from 2 to 5; for the AVR test we compute 500 bootstrap iterations; for DL the table reports both the wild-bootstrap p-values of the Cramer von Mises test statistic (cp) and of the Kolmogorov-Smirnov test statistic (kp). Panel A refers to Bitcoin returns, while Panel B reports the results for the corresponding volatility computed as the absolute value of the returns (i.e., |returns|).

PANEL A | |||||||

Period | RunsTest | BartelsTest | BDSTest | AutomaticPortmanteauTest | AVRTest | DL (cp)Test | DL (kp)Test |

2015/01/01–2015/12/31 | 0.00053 | 0.00005 | 0.00000 | 0.10644 | 0.35000 | 0.00000 | |

2016/01/01–2016/12/31 | 0.01605 | 0.00016 | 0.00000 | 0.08296 | 0.04800 | 0.00000 | 0.00000 |

2017/01/01–2017/12/31 | 0.00164 | 0.00000 | 0.00000 | 0.00041 | 0.00000 | 0.00000 | 0.00000 |

2018/01/01–2018/12/31 | 0.07434 | 0.00070 | 0.00000 | 0.02037 | 0.01400 | 0.00000 | 0.00000 |

2019/01/01–2019/12/31 | 0.00078 | 0.00000 | 0.00148 | 0.00033 | 0.00800 | 0.00000 | 0.00000 |

2018/02/12–2020/08/12 | 0.00002 | 0.00000 | 0.00000 | 0.00002 | 0.00200 | 0.00000 | 0.00000 |

2015/01/01–2020/08/12 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

PANEL B | |||||||

Period | RunsTest | BartelsTest | BDSTest | AutomaticPortmanteauTest | AVRTest | DL (cp)Test | DL (kp)Test |

2015/01/01–2015/12/31 | 0.00036 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2016/01/01–2016/12/31 | 0.02127 | 0.00001 | 0.00000 | 0.00055 | 0.00000 | 0.00000 | 0.00000 |

2017/01/01–2017/12/31 | 0.00016 | 0.00000 | 0.00037 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2018/01/01–2018/12/31 | 0.00000 | 0.00000 | 0.00081 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2019/01/01–2019/12/31 | 0.00016 | 0.00000 | 0.03740 | 0.00092 | 0.00000 | 0.00000 | 0.00000 |

2018/02/12–2020/08/12 | 0.00000 | 0.00000 | 0.00009 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

2015/01/01–2020/08/12 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 0.00000 |

**Table 3.**Testing for granger-causality. The table reports the results of the Toda–Yamamoto test (Toda and Yamamoto 1995). In Panel A we test whether column variables Granger-cause the row variable, while the opposite for Panel B.

Panel A: Column G-Causes Row | BTC Alpha | BTC Vol Alpha | BTC Price | BTC Returns | |

LN efficiency | statistics | 5.50 | 0.70 | 1.70 | 1.60 |

p-value | 0.36 | 0.98 | 0.42 | 0.46 | |

Panel B: Row G-Causes Column | BTC Alpha | BTC Vol Alpha | BTC Price | BTC Returns | |

LN efficiency | statistics | 7.40 | 2.90 | 0.41 | 0.31 |

p-value | 0.19 | 0.72 | 0.81 | 0.86 |

**Table 4.**Testing for the Granger-causality relationship: BTC returns vs. LN configuration. The table reports the results of the Toda–Yamamoto test (Toda and Yamamoto 1995) in which BTC returns are tested to verify whether they Granger-cause a list of topological indicators for the LN (reported in column). These topological indicators refer to respectively: the assortativity, the density, the transitivity, the median value of the nodes’ strength, and the median capacity of the edges.

Row G-Causes Column | Assortativity | Density | Transitivity | Median Strength | Median Capacity | |
---|---|---|---|---|---|---|

BTC returns | statistics | 0.17 | 0.96 | 3.10 | 0.69 | 4.70 |

p-value | 0.68 | 0.33 | 0.21 | 0.41 | 0.03 |

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Martinazzi, S.; Regoli, D.; Flori, A. A Tale of Two Layers: The Mutual Relationship between Bitcoin and Lightning Network. *Risks* **2020**, *8*, 129.
https://doi.org/10.3390/risks8040129

**AMA Style**

Martinazzi S, Regoli D, Flori A. A Tale of Two Layers: The Mutual Relationship between Bitcoin and Lightning Network. *Risks*. 2020; 8(4):129.
https://doi.org/10.3390/risks8040129

**Chicago/Turabian Style**

Martinazzi, Stefano, Daniele Regoli, and Andrea Flori. 2020. "A Tale of Two Layers: The Mutual Relationship between Bitcoin and Lightning Network" *Risks* 8, no. 4: 129.
https://doi.org/10.3390/risks8040129