# Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### Example of Application

- $\alpha <0$
- when the correlation strength is smaller than zero—the distance between time series is decreasing, the time series are converging.
- $\alpha >0$
- when the correlation strength is greater than zero—the distance between time series is increasing, and the time series are diverging.

- converging time series networks, i.e., only the nodes (representing the currency time series) with a correlation strength smaller than one are connected, and
- diverging time series network, i.e., only the nodes with a correlation strength greater than one are connected.

- Clique size evolution is obtained by calculating the size of the biggest clique for each of the generated networks. The clique size evolution illustrates a process of unification of the market. Indeed, if the giant clique is observed, then one type of correlation is dominating on the market and, on the contrary, if the size of the biggest cluster is small, then the correlation matrix consists of a variety of correlation type.
- Community number is obtained by measuring the number of community structure partitions that group nodes, such that there is a higher density of edges within the community than between them. This parameter is weaker than the clique number, but still allows observing grouping on Forex market.
- The frequency of connection on the graph is the measure where the frequency of being connected on the graph is analysed. The most important feature of this measure is the ability to distinguish the most stable connections in the considered period.
- Node rank distribution is the analysis where the most detailed information regarding the graph is obtained. The rank of nodes is an important feature allowing for observing the hierarchy of a network and is often used to determine network type [49,50,51]. This measure gives very detailed information regarding the graph. It may be considered as a quick overview of the network main features, e.g., if it is densely connected or whether each node is only connected with a small number of links.
- Rank node entropy is the Shannon entropy that is defined in the standard way (Equation (1)), where the evolution of the entropy of node rank is calculated.$$S=\sum _{i}-{p}_{i}ln{p}_{i},$$

## 3. Data

#### 3.1. Data Source

#### 3.2. Descriptive Statistics of the Series

## 4. Results

#### 4.1. Month Time Window

#### 4.2. Quarter Time Window

#### 4.3. Half Year Time Window

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 2.**The frequency of connection presented in descending order. The time window ${T}_{c}=20$ days. The blue line denotes a group of currencies of similar frequency of being connected on the network.

**Figure 5.**Evolution of the rank nodes histogram for converging network. The time window size ${T}_{c}=20$ days. The counts denote how many times the node of given rank (number of links) was observed on the network.

**Figure 6.**Evolution of the rank nodes histogram for diverging network. The time window size ${T}_{c}=20$ days. Counts denotes how many times the node of given rank (number of links) was observed on the network.

**Figure 7.**Evolution of the rank node entropy for diverging and converging networks. The time window size ${T}_{c}=20$ days. The blue circles and green squares denote the entropy of diverging and convergent network, respectively.

**Figure 8.**The frequency of connection presented in descending order. The time window ${T}_{c}=60$ days. The blue line denotes group of currencies of similar frequency of being connected on the network.

**Figure 11.**Evolution of the rank nodes histogram for converging network. The time window size ${T}_{c}=60$ days. Counts denote how many times the node of given rank (number of links) was observed on the network.

**Figure 12.**Evolution of the rank nodes histogram for diverging network. The time window size ${T}_{c}=60$ days. Counts denote how many times the node of given rank (number of links) was observed on the network.

**Figure 13.**Evolution of the rank node entropy for diverging and converging networks. The time window size ${T}_{c}=60$ days. The blue circles and green squares denote the entropy of diverging and convergent network respectively.

**Figure 14.**The frequency of connection presented in descending order. The time window ${T}_{c}=120$ days. The blue line denotes group of currencies of similar frequency of being connected on the network.

**Figure 17.**Evolution of the rank nodes histogram for converging network. The time window size ${T}_{c}=120$ days. Counts denotes how many times the node of given rank (number of links) was observed on the network.

**Figure 18.**Evolution of the rank nodes histogram for diverging network. The time window size ${T}_{c}=120$ days. Counts denote how many times the node of given rank (number of links) was observed on the network.

**Figure 19.**Evolution of the rank node entropy for diverging and converging networks. The time window size ${T}_{c}=120$ days. The blue circles and green squares denote the entropy of diverging and convergent network respectively.

Currency | Mean | Median | Std | Max | Min | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

$\xb7{\mathbf{10}}^{-\mathbf{4}}$ | $\xb7{\mathbf{10}}^{-\mathbf{4}}$ | $\xb7{\mathbf{10}}^{-\mathbf{2}}$ | |||||

AR | 9.087 | 3.836 | 1.413 | 0.403 | −0.126 | 11.147 | 261.7 |

CZK | −0.471 | −0.740 | 0.486 | 0.093 | −0.064 | 1.291 | 41.6 |

AUD | 0.324 | −2.245 | 0.760 | 0.079 | −0.050 | 0.743 | 10.8 |

DKK | 0.053 | 0 | 0.048 | 0.079 | −0.009 | −0.560 | 84.1 |

BGN | 1.393 | 0.452 | 0.845 | 0.063 | −0.060 | 0.324 | 6.4 |

EGP | 3.513 | 0.665 | 1.261 | 0.586 | −0.075 | 21.336 | 961.4 |

BRL | 3.334 | −0.718 | 1.182 | 0.129 | −0.1108 | 0.513 | 15.8 |

HKD | −0.079 | 0 | 0.663 | 0.055 | −0.070 | −0.094 | 8.4 |

CAD | −0.140 | −1.480 | 0.674 | 0.044 | −0.043 | 0.201 | 5.7 |

HRK | 0.351 | 0.135 | 0.492 | 0.049 | −0.053 | 0.092 | 18.2 |

CHF | −0.629 | 0 | 0.468 | 0.088 | −0.159 | −6.186 | 304.3 |

HUF | 1.335 | 0.289 | 0.595 | 0.070 | −0.062 | 1.174 | 20.1 |

IDR | 4.849 | 0 | 1.802 | 0.462 | −0.207 | 5.287 | 134.0 |

CNY | −0.313 | 0.329 | 0.834 | 0.050 | −0.062 | −0.102 | 8.4 |

ISK | 1.392 | −0.991 | 0.876 | 0.145 | −0.133 | 1.199 | 71.2 |

JPY | 0.061 | 2.313 | 0.849 | 0.083 | −0.116 | −0.606 | 17.0 |

KRW | 1.050 | −1.545 | 1.084 | 0.158 | −0.232 | −0.678 | 78.1 |

MXN | 1.944 | 0 | 0.904 | 0.068 | −0.091 | 0.221 | 10.1 |

MYR | 1.033 | −0.253 | 0.773 | 0.068 | −0.070 | 0.129 | 13.0 |

NAD | 2.782 | −0.375 | 1.100 | 0.184 | −0.101 | 1.500 | 25.9 |

NOK | 0.585 | −0.937 | 0.530 | 0.050 | −0.082 | −0.350 | 23.5 |

NZD | 0.162 | −3.306 | 0.783 | 0.057 | −0.051 | 0.341 | 6.3 |

PHP | 1.367 | 1.125 | 0.799 | 0.111 | −0.130 | −0.039 | 29.5 |

PLN | 0.615 | −1.442 | 0.642 | 0.057 | −0.048 | 0.609 | 9.6 |

RON | 5.452 | 0.675 | 0.908 | 0.192 | −0.096 | 3.521 | 76.2 |

RUB | 6.074 | 1.393 | 1.651 | 0.347 | −0.282 | 4.050 | 124.6 |

SEK | 0.570 | −0.450 | 0.475 | 0.036 | −0.039 | 0.228 | 8.4 |

SGD | −0.159 | 0 | 0.595 | 0.043 | −0.052 | −0.154 | 7.0 |

THB | 0.412 | 0.367 | 1.016 | 0.171 | −0.067 | 1.045 | 25.0 |

TRY | 9.005 | 4.949 | 1.177 | 0.267 | −0.086 | 4.395 | 89.3 |

TWD | 0.09 | −0.232 | 0.654 | 0.068 | −0.069 | 0.079 | 9.6 |

UAH | 6.469 | 0 | 1.732 | 0.554 | −0.215 | 8.250 | 258.0 |

USD | −0.082 | 0 | 0.672 | 0.077 | −0.077 | −0.046 | 11.6 |

ZAR | 2.768 | −1.856 | 1.113 | 0.121 | −0.143 | 0.259 | 18.1 |

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**MDPI and ACS Style**

Miśkiewicz, J.
Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market. *Entropy* **2021**, *23*, 352.
https://doi.org/10.3390/e23030352

**AMA Style**

Miśkiewicz J.
Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market. *Entropy*. 2021; 23(3):352.
https://doi.org/10.3390/e23030352

**Chicago/Turabian Style**

Miśkiewicz, Janusz.
2021. "Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market" *Entropy* 23, no. 3: 352.
https://doi.org/10.3390/e23030352