# Thermodynamic Analysis of Time Evolving Networks

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## Abstract

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## 1. Introduction

#### Related Literature

## 2. Thermodynamic Framework for Time-Evolving Complex Networks

#### 2.1. Initial Considerations

#### 2.2. Approximate von Neumann Entropy for Undirected Graphs

#### 2.3. Internal Energy and Temperature

#### 2.3.1. Undirected Edges

#### 2.3.2. Directed Edges

#### 2.4. Section Summary

## 3. Experiments and Evaluations

#### 3.1. Thermodynamic Measures for Analysing Network Evolution

#### 3.1.1. Financial Networks

#### 3.1.2. Gene Regulatory Network

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**3D scatter plot of the dynamic stock correlation network in the thermodynamic space. Red dots: 1987–1999 data; cyan dots: Dot-com Bubble; blue dots: 2003–2006 background data; green dots: Subprime Crisis.

**Figure 2.**Top to bottom: (

**a**) the von Neumann entropy versus time for the dynamic stock correlation network; (

**b**) the temperature versus time for the dynamic stock correlation network; (

**c**) the internal energy versus time for the dynamic stock correlation network.

**Figure 3.**Top to bottom: (

**a**) the Estrada index versus time for the dynamic stock correlation network; (

**b**) the assortativity coefficient versus time for the dynamic stock correlation network.

**Figure 4.**Trace of the time-evolving stock correlation network in the entropy-energy plane during financial crises (the number beside the data point is the day number in the time series).

**Left**: Black Monday (from Days 30–300);

**Middle**: Asian Financial Crisis (from Days 2500–2800);

**Right**: Bankruptcy of Lehman Brothers (from days 5300–5500).

**Figure 6.**Scatter plots of variance of ${\Delta}_{v}$ versus ${d}_{u}$ for high and low temperature networks.

**Figure 7.**3D scatter plot of the Drosophila melanogaster gene regulatory network in the thermodynamic space. Red dots: embryonic period; cyan dots: larval period; blue dots: pupal period: green dots: adulthood; black dot: adult ready to emerge.

**Figure 8.**Top to bottom: (

**a**) the von Neumann entropy versus time for the Drosophila melanogaster gene regulatory network; (

**b**) the temperature versus time for the Drosophila melanogaster gene regulatory network; (

**c**) the internal energy versus time for the Drosophila melanogaster gene regulatory network.

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**MDPI and ACS Style**

Ye, C.; Wilson, R.C.; Rossi, L.; Torsello, A.; Hancock, E.R.
Thermodynamic Analysis of Time Evolving Networks. *Entropy* **2018**, *20*, 759.
https://doi.org/10.3390/e20100759

**AMA Style**

Ye C, Wilson RC, Rossi L, Torsello A, Hancock ER.
Thermodynamic Analysis of Time Evolving Networks. *Entropy*. 2018; 20(10):759.
https://doi.org/10.3390/e20100759

**Chicago/Turabian Style**

Ye, Cheng, Richard C. Wilson, Luca Rossi, Andrea Torsello, and Edwin R. Hancock.
2018. "Thermodynamic Analysis of Time Evolving Networks" *Entropy* 20, no. 10: 759.
https://doi.org/10.3390/e20100759