# Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Simplicial Complex in the Context of Case Study

#### 2.2. Multilevel Integration Entropies

#### 2.3. Results of the Calculations

## 3. Discussion

## 4. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Simplicial Complex

#### Appendix A.2. Data Preparation

- (1)
- appearance of zero values at the places of latitude and longitude coordinates;
- (2)
- the recordings of some rides were repeating;
- (3)
- the latitude and longitude coordinates of some rides are (far) out of the city border.

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**Figure 1.**An example of simplicial complex construction from the taxi driver dataset when the city is divided into the grid of cells in the case when the taxi driver picked up passengers at the location within the cells i and k and dropped them off at destinations ${j}_{1}$, ${j}_{2}$, ${j}_{3}$, ${j}_{4}$ and ${j}_{2}$, ${j}_{4}$, ${j}_{5}$, respectively.

**Figure 2.**An example of updating the simplicial complex of origins when new rides of the taxi driver are added in two consecutive time moments ($t+1$ and $t+2$) from some arbitrary moment t. The origins and the associated simplices are labeled by the letters, whereas the destinations and the associated vertices are labeled by the numerals.

**Figure 3.**An example of the embeddedness of a connected collection of simplices in simplicial complex (

**a**) and a simplex in the connectivity class (

**b**).

**Figure 4.**Time change of the values of the $\mathbf{H}$ entropy entries for simplicial complex of origins (

**a**) and its conjugate complex (

**b**).

**Figure 5.**Time change of the values of $\mathbf{HQ}$ entropy entries for simplicial complex of origins (

**a**) and its conjugate complex (

**b**).

**Figure 6.**Entropy structural coefficients $\chi $ (

**a**–

**c**) and $\theta $ (

**d**–

**f**) for simplicial complex and its conjugate for the original data (

**a**,

**d**), for the randomized destinations (

**b**,

**e**), and for the randomized origins (

**c**,

**e**) under the transition $\tau =t\to t+1$.

**Figure 7.**Time change of the maximum values of (

**a**) $\mathbf{H}$ and (

**b**) $\mathbf{HQ}$ entropy entries for simplicial complex of origins and its conjugate complex of the original data.

**Figure 8.**Time change of the maximum values of (

**a**) $\mathbf{H}$ and (

**b**) $\mathbf{HQ}$ entropy entries for simplicial complex of origins and its conjugate complex for the randomized destinations.

**Figure 9.**Time change of the maximum values of (

**a**) $\mathbf{H}$ and (

**b**) $\mathbf{HQ}$ entropy entries for simplicial complex of origins and its conjugate complex for the randomized origins.

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**MDPI and ACS Style**

Maletić, S.; Zhao, Y.
Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets. *Entropy* **2017**, *19*, 172.
https://doi.org/10.3390/e19040172

**AMA Style**

Maletić S, Zhao Y.
Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets. *Entropy*. 2017; 19(4):172.
https://doi.org/10.3390/e19040172

**Chicago/Turabian Style**

Maletić, Slobodan, and Yi Zhao.
2017. "Multilevel Integration Entropies: The Case of Reconstruction of Structural Quasi-Stability in Building Complex Datasets" *Entropy* 19, no. 4: 172.
https://doi.org/10.3390/e19040172