# Structural Correlations in the Italian Overnight Money Market: An Analysis Based on Network Configuration Models

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## Abstract

**:**

## 1. Introduction

## 2. Structural Correlations and Configuration Models

#### 2.1. Undirected Networks

#### 2.1.1. General Notation

#### 2.1.2. Structural Correlations in Undirected Networks

#### Assortativity Analysis

#### The Average Degree and Strength of the Nearest Neighbors

#### Pearson Correlation Coefficient of the Node Degrees and of the Node Strengths

#### Clustering Coefficients

#### 2.2. Directed Networks

#### 2.2.1. General Definitions

#### 2.2.2. Structural Correlations in Directed Networks

#### Assortativity Analysis

#### Clustering Coefficients

#### 2.3. Configuration Models

#### Undirected Binary Configuration Model (UBCM)

#### Directed Binary Configuration Model (DBCM)

#### Undirected Weighted Configuration Model (UWCM)

#### Directed Weighted Configuration Model (DWCM)

#### Undirected Enhanced Configuration Model (UECM)

#### Directed Enhanced Configuration Model (DECM)

## 3. Data and Summary Statistics for the Italian Interbank Market e-MID

## 4. Findings for the Binary Network

#### 4.1. Structural Correlations in the Undirected Binary e-MID Network

#### 4.2. Structural Correlations in the Directed Binary e-MID Network

#### 4.3. Comparisons to the Configuration Models

#### 4.3.1. Undirected Binary Network

#### 4.3.2. Directed Binary Network

## 5. Findings for the Weighted Network

#### 5.1. Structural Correlations in the Undirected Weighted e-MID Network

#### 5.2. Structural Correlations in the Directed Weighted e-MID Network

#### 5.3. Comparisons to the Weighted Configuration Models

#### 5.3.1. Undirected Weighted Network

#### 5.3.2. Directed Weighted Network

#### 5.4. z-Scores Analysis Revealing Structural Changes in the Weighted System

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Overall Assortativity

#### Appendix A.2. Local Assortativity

## Appendix B

**Figure A2.**Histogram of expected undirected link probabilities, $\langle {a}_{ij}\rangle $, under the UBCM in Q1 and Q48.

**Figure A3.**Histogram of expected directed link probabilities, $\langle {a}_{ij}\rangle $, under the DBCM in Q1 and Q48.

**Figure A4.**Histogram of expected undirected link probabilities, $\langle {a}_{ij}\rangle $, under the UWCM in Q1 and Q48.

**Figure A5.**Histogram of expected undirected weights, $\langle {w}_{ij}\rangle $, under the UWCM in Q1 and Q48. Note that for the sake of readability, we plot the number of the observations in log scale.

**Figure A6.**Histogram of expected undirected link probabilities, $\langle {a}_{ij}\rangle $, under the UECM in Q1 and Q48.

**Figure A7.**Histogram of expected undirected weights, $\langle {w}_{ij}\rangle $, under the UECM in Q1 and Q48. Note that for the sake of readability, we plot the number of the observations in log scale.

**Figure A8.**Histogram of expected directed link probabilities, $\langle {a}_{ij}\rangle $, under the DWCM in Q1 and Q48.

**Figure A9.**Histogram of expected directed weights, $\langle {w}_{ij}\rangle $, under the DWCM in Q1 and Q48. Note that for the sake of readability, we plot the number of the observations in log scale.

**Figure A10.**Histogram of expected directed link probabilities, $\langle {a}_{ij}\rangle $, under the DECM in Q1 and Q48.

**Figure A11.**Histogram of expected directed weights, $\langle {w}_{ij}\rangle $, under the DECM in Q1 and Q48. Note that for the sake of readability, we plot the number of the observations in log scale.

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**Figure 1.**Degree-degree dependencies in the directed version.Degree-degree dependencies in the directed version.

**Figure 2.**Directed triangles and the corresponding (binary) clusterings associated with a node i. (

**a**) Cycle clustering; (

**b**) Middleman clustering; (

**c**) In clustering; (

**d**) Out clustering.Directed triangles and the corresponding clusterings.

**Figure 3.**The evolution of the size and the density of the electronic market for interbank deposits (e-MID) network. The left y-axis shows the number of active banks (in blue) and the right y-axis shows density (in green).

**Figure 4.**The evolution of the basic statistics of weights in the undirected version (${w}^{un}$) and in directed version (${w}^{dir}$).

**Figure 5.**Histograms of weights (i.e., ${w}_{ij}^{un}$ in the undirected version and ${w}_{ij}^{dir}$ in the directed version) in two example quarters, Q1 and Q48. Note that for the sake of readability, we plot the number of the observations in log scale.

**Figure 6.**The evolution of the basic statistics of degrees. Note that, by definition, the mean values of ${k}^{in}$ and ${k}^{out}$ are identical.

**Figure 8.**The evolution of the basic statistics of strengths. Note that, by definition, the mean values of ${s}^{in}$ and ${s}^{out}$ are identical.

**Figure 11.**Average degree of the nearest neighbors (ANND) (

**a**,

**b**), local assortativity ${\rho}^{un}$ (

**c**,

**d**), and local clustering coefficients ${C}_{bin}^{un}$ (

**e**,

**f**) in the undirected binary e-MID network, in Q1 and Q48.Average degrees of nearest neighbors, local assortativity, and local clustering coefficients in the undirected binary e-MID network, in Q1 and Q48.

**Figure 12.**Evolution of the overall assortativity indicator ${r}_{bin}^{un}$ in the undirected binary e-MID network.

**Figure 13.**Evolution of the average of local clustering coefficients (i.e., ${\overline{C}}_{bin}^{un}$) in the undirected binary e-MID network.

**Figure 14.**ANND in the directed binary e-MID network, in Q1. (

**a**) ${k}_{nn}^{in-in}$; (

**b**) ${k}_{nn}^{in-out}$; (

**c**) ${k}_{nn}^{out-in}$; (

**d**) ${k}_{nn}^{out-out}$.

**Figure 15.**ANND in the directed binary e-MID network, in Q48. (

**a**) ${k}_{nn}^{in-in}$; (

**b**) ${k}_{nn}^{in-out}$; (

**c**) ${k}_{nn}^{out-in}$; (

**d**) ${k}_{nn}^{out-out}$.

**Figure 16.**Evolution of the overall assortativity indicators in the directed binary e-MID network.Evolution of the overall assortativity indicators in the directed binary e-MID network.

**Figure 17.**Local assortativity in the directed binary e-MID network, in Q1. (

**a**) ${\rho}^{in-in}$; (

**b**) ${\rho}^{in-out}$; (

**c**) ${\rho}^{out-in}$; (

**d**) ${\rho}^{out-out}$.

**Figure 18.**Local assortativity in the directed binary e-MID network, in Q48. (

**a**) ${\rho}^{in-in}$; (

**b**) ${\rho}^{in-out}$; (

**c**) ${\rho}^{out-in}$; (

**d**) ${\rho}^{out-out}$.

**Figure 19.**Local clustering coefficients (

**a**) ${C}_{bin}^{cyc}$; (

**b**) ${C}_{bin}^{mid}$; (

**c**) ${C}_{bin}^{in}$; (

**d**) ${C}_{bin}^{out}$ in the directed binary e-MID network, in Q1.

**Figure 20.**Local clustering coefficients (

**a**) ${C}_{bin}^{cyc}$; (

**b**) ${C}_{bin}^{mid}$; (

**c**) ${C}_{bin}^{in}$; (

**d**) ${C}_{bin}^{out}$ in the directed binary e-MID network, in Q48.

**Figure 21.**Evolution of the averages of local clustering coefficients (i.e., ${\overline{C}}^{mid}$, ${\overline{C}}^{in}$, ${\overline{C}}^{out}$, and ${\overline{C}}^{cyc}$) in the directed binary e-MID network.

**Figure 22.**(

**a**,

**b**) ANND; (

**c**,

**d**) local assortativity ${\rho}^{un}$; (

**e**,

**f**) local clustering coefficients ${C}_{bin}^{un}$ in the observed e-MID network and in the UBCM, in Q1 and Q48.

**Figure 23.**Evolution of (

**a**) ${\overline{k}}_{nn}^{un}$; (

**b**) ${r}_{bin}^{un}$; and (

**c**) ${\overline{C}}_{in}^{un}$ in the observed e-MID network and in the undirected binary configuration model (UBCM).

**Figure 24.**ANND in the observed e-MID network and in the DBCM, in Q1. (

**a**) ${k}_{nn}^{in-in}$; (

**b**) ${k}_{nn}^{in-out}$; (

**c**) ${k}_{nn}^{out-in}$; (

**d**) ${k}_{nn}^{out-out}$.

**Figure 25.**ANND in the observed e-MID network and in the DBCM, in Q48. (

**a**) ${k}_{nn}^{in-in}$; (

**b**) ${k}_{nn}^{in-out}$; (

**c**) ${k}_{nn}^{out-in}$; (

**d**) ${k}_{nn}^{out-out}$.

**Figure 26.**Local assortativity in the observed e-MID network and in the DBCM, in Q1. (

**a**) ${\rho}^{in-in}$; (

**b**) ${\rho}^{in-out}$; (

**c**) ${\rho}^{out-in}$; (

**d**) ${\rho}^{out-out}$.

**Figure 27.**Local assortativity in the observed e-MID network and in the DBCM, in Q48. (

**a**) ${\rho}^{in-in}$; (

**b**) ${\rho}^{in-out}$; (

**c**) ${\rho}^{out-in}$; (

**d**) ${\rho}^{out-out}$.

**Figure 28.**Local clustering coefficients (

**a**) ${C}_{bin}^{cyc}$; (

**b**) ${C}_{bin}^{mid}$; (

**c**) ${C}_{bin}^{in}$; (

**d**) ${C}_{bin}^{out}$ in the observed e-MID network and in DBCM, in Q1.

**Figure 29.**Local clustering coefficients (

**a**) ${C}_{bin}^{cyc}$; (

**b**) ${C}_{bin}^{mid}$; (

**c**) ${C}_{bin}^{in}$; (

**d**) ${C}_{bin}^{out}$ in the observed e-MID network and in DBCM, in Q48.

**Figure 30.**Evolution of the averages of ANNDs in the observed e-MID network and in the DBCM. (

**a**) ${\overline{k}}_{nn}^{in-in}$; (

**b**) ${\overline{k}}_{nn}^{in-out}$; (

**c**) ${\overline{k}}_{nn}^{out-in}$; (

**d**) ${\overline{k}}_{nn}^{out-out}$.

**Figure 31.**Evolution of the global assortativity indicators in the observed e-MID network and in the DBCM. (

**a**) ${r}_{bin}^{in-in}$; (

**b**) ${r}_{bin}^{in-out}$; (

**c**) ${r}_{bin}^{out-in}$; (

**d**) ${r}_{bin}^{out-out}$.

**Figure 32.**Evolution of the averages of clustering coefficients in the observed e-MID network and in the DBCM. (

**a**) ${\overline{C}}_{bin}^{cyc}$; (

**b**) ${\overline{C}}_{bin}^{mid}$; (

**c**) ${\overline{C}}_{bin}^{in}$; (

**d**) ${\overline{C}}_{bin}^{out}$.

**Figure 33.**Average strength of the nearest neighbors (ANNS) in the undirected weighted e-MID network, in Q1 and Q48.

**Figure 34.**Evolution of global weighted assortativity ${r}_{w}^{un}$ in the undirected weighted e-MID network.

**Figure 35.**Local clustering coefficients ${C}_{w}^{un}$ in the undirected weighted e-MID network, in Q1 and Q48.

**Figure 36.**Evolution of the average of local weighted clustering coefficients (i.e., ${\overline{C}}_{w}^{un}$) in the undirected weighted e-MID network.

**Figure 37.**ANNSs in the directed weighted e-MID network, in Q1. (

**a**) ${s}_{nn}^{in-in}$; (

**b**) ${s}_{nn}^{in-out}$; (

**c**) ${s}_{nn}^{out-in}$; (

**d**) ${s}_{nn}^{out-out}$.

**Figure 38.**ANNSs in the directed weighted e-MID network, in Q48. (

**a**) ${s}_{nn}^{in-in}$; (

**b**) ${s}_{nn}^{in-out}$; (

**c**) ${s}_{nn}^{out-in}$; (

**d**) ${s}_{nn}^{out-out}$.

**Figure 39.**Evolution of the directed weighted assortativity indicators, i.e., ${r}_{w}^{in-in}$, ${r}_{w}^{in-out}$, ${r}_{w}^{out-in}$, and ${r}_{w}^{out-out}$ in the directed weighted e-MID network.

**Figure 40.**Local weighted clustering coefficients in the directed weighted e-MID network, in Q1. (

**a**) ${C}_{w}^{cyc}$; (

**b**) ${C}_{w}^{mid}$; (

**c**) ${C}_{w}^{in}$; (

**d**) ${C}_{w}^{out}$.

**Figure 41.**Local weighted clustering coefficients in the directed weighted e-MID network, in Q48. (

**a**) ${C}_{w}^{cyc}$; (

**b**) ${C}_{w}^{mid}$; (

**c**) ${C}_{w}^{in}$; (

**d**) ${C}_{w}^{out}$.

**Figure 42.**Evolution of the averages of local weighted clustering coefficients, i.e., ${\overline{C}}_{w}^{cyc}$, ${\overline{C}}_{w}^{mid}$, ${\overline{C}}_{w}^{in}$, and ${\overline{C}}_{w}^{out}$ in the directed weighted e-MID network.

**Figure 44.**Local weighted clustering coefficients ${C}_{w}^{un}$ in the observed e-MID network and in the UWCM, in Q1 and Q48.

**Figure 46.**Local weighted clustering coefficients ${C}_{w}^{un}$ in the observed e-MID network and in the UECM, in Q1 and Q48.

**Figure 47.**z-scores of ${s}_{nn}^{un}$ vs. ${s}_{nn}^{un}$ in the UWCM and the UECM, in Q1 and in Q48. Panel (

**a**) for z-scores of ${s}_{nn}^{un}$ in Q1; panel (

**b**) for z-scores of ${s}_{nn}^{un}$ in Q48.

**Figure 48.**z-scores of ${C}_{w}^{un}$ vs. ${C}_{w}^{un}$ in the UWCM and the UECM. (

**a**) for z-scores of ${C}_{w}^{un}$ in Q1; (

**b**) for z-scores of ${C}_{w}^{un}$ in Q48.

**Figure 49.**Evolution of (

**a**) ${\overline{s}}_{nn}^{un}$; (

**b**) ${r}_{w}^{un}$; and (

**c**) ${\overline{C}}_{w}^{un}$ in the observed e-MID network and in the UWCM.

**Figure 50.**Evolution of (

**a**) ${\overline{s}}_{nn}^{un}$; (

**b**) ${r}_{w}^{un}$; and (

**c**) ${\overline{C}}_{w}^{un}$ in the observed e-MID network and in the UECM.

**Figure 51.**ANNSs in the observed e-MID network and in the DWCM, in Q1. (

**a**) ${s}_{nn}^{in-in}$; (

**b**) ${s}_{nn}^{in-out}$; (

**c**) ${s}_{nn}^{out-in}$; (

**d**) ${s}_{nn}^{out-out}$.

**Figure 52.**ANNSs in the observed e-MID network and in the DWCM, in Q48. (

**a**) ${s}_{nn}^{in-in}$; (

**b**) ${s}_{nn}^{in-out}$; (

**c**) ${s}_{nn}^{out-in}$; (

**d**) ${s}_{nn}^{out-out}$.

**Figure 53.**ANNSs in the observed e-MID network and in the DECM, in Q1. (

**a**) ${s}_{nn}^{in-in}$; (

**b**) ${s}_{nn}^{in-out}$; (

**c**) ${s}_{nn}^{out-in}$; (

**d**) ${s}_{nn}^{out-out}$.

**Figure 54.**ANNSs in the observed e-MID network and in the DECM, in Q48. (

**a**) ${s}_{nn}^{in-in}$; (

**b**) ${s}_{nn}^{in-out}$; (

**c**) ${s}_{nn}^{out-in}$; (

**d**) ${s}_{nn}^{out-out}$.

**Figure 55.**z-scores of ANNSs vs. ANNSs, in the DWCM and DECM models, in Q1. (

**a**) for ${s}_{nn}^{in-in}$; (

**b**) for ${s}_{nn}^{in-out}$; (

**c**) for ${s}_{nn}^{out-in}$; (

**d**) for ${s}_{nn}^{out-out}$.

**Figure 56.**z-scores of ANNSs vs. ANNSs, in the DWCM and DECM models, in Q48. (

**a**) for ${s}_{nn}^{in-in}$; (

**b**) for ${s}_{nn}^{in-out}$; (

**c**) for ${s}_{nn}^{out-in}$; (

**d**) for ${s}_{nn}^{out-out}$.

**Figure 57.**Local weighted clustering coefficients in the observed e-MID network and in the DWCM, in Q1. (

**a**) ${C}_{w}^{cyc}$; (

**b**) ${C}_{w}^{mid}$; (

**c**) ${C}_{w}^{in}$; (

**d**) ${C}_{w}^{out}$.

**Figure 58.**Local weighted clustering coefficients in the observed e-MID network and in the DWCM, in Q48. (

**a**) ${C}_{w}^{cyc}$; (

**b**) ${C}_{w}^{mid}$; (

**c**) ${C}_{w}^{in}$; (

**d**) ${C}_{w}^{out}$.

**Figure 59.**Local weighted clustering coefficients in the observed e-MID network and in the DECM, in Q1. (

**a**) ${C}_{w}^{cyc}$; (

**b**) ${C}_{w}^{mid}$; (

**c**) ${C}_{w}^{in}$; (

**d**) ${C}_{w}^{out}$.

**Figure 60.**Local weighted clustering coefficients in the observed e-MID network and in the DECM, in Q48. (

**a**) ${C}_{w}^{cyc}$; (

**b**) ${C}_{w}^{mid}$; (

**c**) ${C}_{w}^{in}$; (

**d**) ${C}_{w}^{out}$.

**Figure 61.**z-scores of ${C}_{w}$ vs. ${C}_{w}$, evaluated under the DWCM and DECM models, in Q1. (

**a**) for ${C}_{w}^{cyc}$; (

**b**) for ${C}_{w}^{mid}$; (

**c**) for ${C}_{w}^{in}$; (

**d**) for ${C}_{w}^{out}$.

**Figure 62.**z-scores of ${C}_{w}$ vs. ${C}_{w}$, evaluated under the DWCM and DECM models, in Q48. (

**a**) for ${C}_{w}^{cyc}$; (

**b**) for ${C}_{w}^{mid}$; (

**c**) for ${C}_{w}^{in}$; (

**d**) for ${C}_{w}^{out}$.

**Figure 63.**Evolution of the averages of ANNSs in the observed e-MID network and in the DWCM. (

**a**) ${\overline{s}}_{nn}^{in-in}$; (

**b**) ${\overline{s}}_{nn}^{in-out}$; (

**c**) ${\overline{s}}_{nn}^{out-in}$; (

**d**) ${\overline{s}}_{nn}^{out-out}$.

**Figure 64.**Evolution of the global weighted assortativity indicators in the observed e-MID network and in the DWCM. (

**a**) ${r}_{w}^{in-in}$; (

**b**) ${r}_{w}^{in-out}$; (

**c**) ${r}_{w}^{out-in}$; (

**d**) ${r}_{w}^{out-out}$.

**Figure 65.**Evolution of the averages of local weighted clustering coefficients in the observed e-MID network and in the DWCM. (

**a**) ${\overline{C}}_{w}^{cyc}$; (

**b**) ${\overline{C}}_{w}^{mid}$; (

**c**) ${\overline{C}}_{w}^{in}$; (

**d**) ${\overline{C}}_{w}^{out}$.

**Figure 66.**Evolution of the averages of ANNSs in the observed e-MID network and in the DECM. (

**a**) ${\overline{s}}_{nn}^{in-in}$; (

**b**) ${\overline{s}}_{nn}^{in-out}$; (

**c**) ${\overline{s}}_{nn}^{out-in}$; (

**d**) ${\overline{s}}_{nn}^{out-out}$.

**Figure 67.**Evolution of the global weighted assortativity indicators in the observed e-MID network and in the DECM. (

**a**) ${r}_{w}^{in-in}$; (

**b**) ${r}_{w}^{in-out}$ (

**b**); (

**c**) ${r}_{w}^{out-in}$; (

**d**) ${r}_{w}^{out-out}$.

**Figure 68.**Evolution of the averages of local weighted clustering coefficients in the observed e-MID network and in the DECM. (

**a**) ${\overline{C}}_{w}^{cyc}$; (

**b**) ${\overline{C}}_{w}^{mid}$; (

**c**) ${\overline{C}}_{w}^{in}$; (

**d**) ${\overline{C}}_{w}^{out}$.

**Figure 69.**Evolution of the z-scores for (

**a**) ${\overline{s}}_{nn}^{un}$; (

**b**) ${r}_{w}^{un}$; and (

**c**) ${\overline{C}}_{w}^{un}$ evaluated under the UWCM (red dashed lines) and the UECM (blue dashed lines).

**Figure 70.**Evolution of the z-scores for (

**a**) ${r}_{w}^{in-in}$; (

**b**) ${r}_{w}^{in-out}$; (

**c**) ${r}_{w}^{out-in}$, and (

**d**) ${r}_{w}^{out-out}$ evaluated under the DWCM (red dashed lines) and the DECM (blue dashed lines).

**Figure 71.**Evolution of the z-scores for (

**a**) ${\overline{s}}_{nn}^{in-in}$; (

**b**) ${\overline{s}}_{nn}^{in-out}$; (

**c**) ${\overline{s}}_{nn}^{out-in}$; and (

**d**) ${\overline{s}}_{nn}^{out-out}$ evaluated under the DWCM (red dashed lines) and the DECM (blue dashed lines).

**Figure 72.**Evolution of the z-scores for (

**a**) ${\overline{C}}_{w}^{cyc}$; (

**b**) ${\overline{C}}_{w}^{mid}$; (

**c**) ${\overline{C}}_{w}^{in}$; and (

**d**) ${\overline{C}}_{w}^{out}$ evaluated under the DWCM (red dashed lines) and the DECM (blue dashed lines).

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Luu, D.T.; Lux, T.; Yanovski, B.
Structural Correlations in the Italian Overnight Money Market: An Analysis Based on Network Configuration Models. *Entropy* **2017**, *19*, 259.
https://doi.org/10.3390/e19060259

**AMA Style**

Luu DT, Lux T, Yanovski B.
Structural Correlations in the Italian Overnight Money Market: An Analysis Based on Network Configuration Models. *Entropy*. 2017; 19(6):259.
https://doi.org/10.3390/e19060259

**Chicago/Turabian Style**

Luu, Duc Thi, Thomas Lux, and Boyan Yanovski.
2017. "Structural Correlations in the Italian Overnight Money Market: An Analysis Based on Network Configuration Models" *Entropy* 19, no. 6: 259.
https://doi.org/10.3390/e19060259