# Low-Temperature Behaviour of Social and Economic Networks

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## Abstract

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**PACS**89.75.Hc; 89.75.Fb; 05.70.-a

## 1. Introduction

## 2. Temperature-Dependent Ensembles of Graphs

#### 2.1. General Formalism

#### 2.2. Networks with Finite Energy Per Link

## 3. Random Graphs: Vanishing of the Percolation Threshold at Zero Temperature

#### 3.1. Critical Percolation Temperature

#### 3.2. Large and Sparse Graphs Have Low Temperature

## 4. Fitness Models: Random Graphs at High Temperature, Scale-Free Networks at Low Temperature

#### 4.1. High-Temperature Regime ($T=+\infty $)

#### 4.2. Finite-Temperature Regime ($T=1$)

#### 4.3. Zero-Temperature Regime ($T=0$)

#### 4.4. The Temperature of Real Binary Networks

## 5. More General Models

## 6. A Temperature-Driven Small-World Model

#### 6.1. Non-Scale-Free Small-Worlds

**Figure 1.**A temperature-dependent small-world model with vertices arranged in a circle and chemical potential, $d<\mu <2d$ (where d is the dimensionless distance between nearest neighbours along the circle). When $T=0$ (left), the network is a ring with first-neighbour interactions. When $T=\infty $ (right), the network is a random graph with connection probability, $p=1/2$. When $T=1$ (center), the network is a “small-world” with a few long-range connections and an incomplete circular “backbone”.

#### 6.2. Scale-Free Small-Worlds

## 7. A Model of Networks with Low-Temperature Community Structure

#### 7.1. Ultrametric Small-World Model

**Figure 2.**Our “ultrametric small-world model” as a function of temperature, T, and chemical potential, μ. Nodes (blue circles) are leaves of a dendrogram (black lines), separated by an ultrametric distance, ${d}_{ij}$ (increasing along the purple axis), representing the height of the closest branching point separating vertices i and j. The ultrametric distances determine the topology of the network (lying on the horizontal purple plane): (

**a**) when $T=0$ and μ is small, the network is divided into many small cliques (blue links) corresponding to the disconnected branches obtained by “cutting” the dendrogram along the orange dashed line determined by μ; (

**b**) when $T=0$ and μ is larger, the network is divided into fewer and larger cliques; (

**c**) when $T\gtrsim 0$ and μ is small, there are many small communities that are highly connected internally (blue links) and sparsely connected across (red links); (

**d**) when $T\gtrsim 0$ and μ is larger, there are fewer and larger communities, with a higher density contrast between intra-community (blue) and inter-community (red) links. After introducing an appropriate degree of heterogeneity at the level of vertices, this model can be turned into our “ultrametric scale-free model”, where a non-trivial community structure coexists with a broad degree distribution.

#### 7.2. Ultrametric Scale-Free Model

## 8. Weighted Networks as Temperature-Dependent Ensembles of Binary Graphs

#### 8.1. The Temperature of Real Weighted Networks

Network | α | ${T}_{\mathrm{weighted}}$ | Ref. |
---|---|---|---|

Metabolic flux networks | $1.5$ | $0.5$ | [29] |

Interbank network | $1.87$ | $0.87$ | [30] |

Erdős collaboration network | 2 | 1 | [31] |

Chaos control & synchron. co-authorship | $2.5$ | $1.5$ | [31] |

Financial cross-correlations | $2.7$ | $1.7$ | [32] |

Financial cross-correlations | $2.78$ | $1.78$ | [33] |

Financial cross-correlations | $3.18$ | $2.18$ | [33] |

Mollusk research co-authorship | $3.5$ | $2.5$ | [31] |

Binary graphs | $+\infty $ | $+\infty $ |

#### 8.2. Filtering of Weighted Networks as the Zero-Temperature Limit

## 9. Conclusions

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Garlaschelli, D.; Ahnert, S.E.; Fink, T.M.A.; Caldarelli, G. Low-Temperature Behaviour of Social and Economic Networks. *Entropy* **2013**, *15*, 3148-3169.
https://doi.org/10.3390/e15083238

**AMA Style**

Garlaschelli D, Ahnert SE, Fink TMA, Caldarelli G. Low-Temperature Behaviour of Social and Economic Networks. *Entropy*. 2013; 15(8):3148-3169.
https://doi.org/10.3390/e15083238

**Chicago/Turabian Style**

Garlaschelli, Diego, Sebastian E. Ahnert, Thomas M. A. Fink, and Guido Caldarelli. 2013. "Low-Temperature Behaviour of Social and Economic Networks" *Entropy* 15, no. 8: 3148-3169.
https://doi.org/10.3390/e15083238