# Distance Entropy Cartography Characterises Centrality in Complex Networks

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## Abstract

**:**

## 1. Introduction

## 2. Introducing a New Distance Entropy

#### Distance Entropy in Regular Graphs

## 3. Distance Entropy and Network Models

#### 3.1. Homogeneous Random Graphs

#### 3.2. Small-World Networks

#### 3.3. Barabasi–Albert Networks

## 4. Cartography Based on Distance Entropy and Closeness Centrality

## 5. Applying Distance Entropy Cartography to Multiplex Lexical Networks

## 6. Discussion

## Author Contributions

## Conflicts of Interest

## References

- Albert, R.; Barabási, A.L. Statistical mechanics of complex networks. Rev. Mod. Phys.
**2002**, 74, 47–97. [Google Scholar] [CrossRef] - Newman, M. Networks: An Introduction; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- De Domenico, M.; Solé-Ribalta, A.; Cozzo, E.; Kivelä, M.; Moreno, Y.; Porter, M.A.; Gómez, S.; Arenas, A. Mathematical formulation of multilayer networks. Phys. Rev. X
**2013**, 3, 041022. [Google Scholar] [CrossRef] - Gao, Z.; Dang, W.; Mu, C.; Yang, Y.; Li, S.; Grebogi, C. A Novel Multiplex Network-based Sensor Information Fusion Model and Its Application to Industrial Multiphase Flow System. IEEE Trans. Ind. Inform.
**2017**. [Google Scholar] [CrossRef] - Gao, Z.K.; Li, S.; Dang, W.D.; Yang, Y.X.; Do, Y.; Grebogi, C. Wavelet multiresolution complex network for analyzing multivariate nonlinear time series. Int. J. Bifurc. Chaos
**2017**, 27, 1750123. [Google Scholar] [CrossRef] - Stella, M.; Beckage, N.M.; Brede, M.; De Domenico, M. Multiplex model of mental lexicon reveals explosive learning in humans. Sci. Rep.
**2018**, 8, 2259. [Google Scholar] [CrossRef] [PubMed] - Borgatti, S.P. Centrality and network flow. Soc. Netw.
**2005**, 27, 55–71. [Google Scholar] [CrossRef] - Morone, F.; Makse, H.A. Influence maximization in complex networks through optimal percolation. Nature
**2015**, 524, 65–68. [Google Scholar] [CrossRef] [PubMed] - Brede, M.; Stella, M.; Kalloniatis, A.C. Competitive influence maximization and enhancement of synchronization in populations of non-identical Kuramoto oscillators. Sci. Rep.
**2018**, 8, 702. [Google Scholar] [CrossRef] [PubMed] - Buldyrev, S.V.; Parshani, R.; Paul, G.; Stanley, H.E.; Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature
**2010**, 464, 1025–1028. [Google Scholar] [CrossRef] [PubMed] - De Domenico, M.; Arenas, A. Modeling structure and resilience of the dark network. Phys. Rev. E
**2017**, 95, 022313. [Google Scholar] [CrossRef] [PubMed] - Collins, A.M.; Loftus, E.F. A spreading-activation theory of semantic processing. Psychol. Rev.
**1975**, 82, 407–428. [Google Scholar] [CrossRef] - De Domenico, M. Diffusion geometry unravels the emergence of functional clusters in collective phenomena. Phys. Rev. Lett.
**2017**, 118, 168301. [Google Scholar] [CrossRef] [PubMed] - Rouquette, J.R.; Dallimer, M.; Armsworth, P.R.; Gaston, K.J.; Maltby, L.; Warren, P.H. Species turnover and geographic distance in an urban river network. Divers. Distrib.
**2013**, 19, 1429–1439. [Google Scholar] [CrossRef] - Bavelas, A. Communication patterns in task-oriented groups. J. Acoust. Soc. Am.
**1950**, 22, 725–730. [Google Scholar] [CrossRef] - Marchiori, M.; Latora, V. Harmony in the small-world. Phys. A Stat. Mech. Appl.
**2000**, 285, 539–546. [Google Scholar] [CrossRef] [Green Version] - Goldstein, R.; Vitevitch, M.S. The Influence of Closeness Centrality on Lexical Processing. Front. Psychol.
**2017**, 8, 1683. [Google Scholar] [CrossRef] [PubMed] - Stella, M.; Beckage, N.M.; Brede, M. Multiplex lexical networks reveal patterns in early word acquisition in children. Sci. Rep.
**2017**, 7, 46730. [Google Scholar] [CrossRef] [PubMed] - Castro, N.; Stella, M. The multiplex structure of the mental lexicon influences picture naming in people with aphasia. PsyarXiv, 2018. [Google Scholar] [CrossRef]
- Dehmer, M.; Mowshowitz, A. A history of graph entropy measures. Inf. Sci.
**2011**, 181, 57–78. [Google Scholar] [CrossRef] - Chen, Z.; Dehmer, M.; Shi, Y. A note on distance-based graph entropies. Entropy
**2014**, 16, 5416–5427. [Google Scholar] [CrossRef] - Dehmer, M.; Emmert-Streib, F.; Chen, Z.; Li, X.; Shi, Y. Mathematical Foundations and Applications of Graph Entropy; John Wiley & Sons: New York, NY, USA, 2017; Volume 6. [Google Scholar]
- Das, K.; Shi, Y. Some properties on entropies of graphs. MATCH Commun. Math. Comput. Chem.
**2017**, 78, 259–272. [Google Scholar] - Brandes, U. A faster algorithm for betweenness centrality. J. Math. Sociol.
**2001**, 25, 163–177. [Google Scholar] [CrossRef] - Brandes, U. On variants of shortest-path betweenness centrality and their generic computation. Soc. Netw.
**2008**, 30, 136–145. [Google Scholar] [CrossRef] - Opsahl, T.; Agneessens, F.; Skvoretz, J. Node centrality in weighted networks: Generalizing degree and shortest paths. Soc. Netw.
**2010**, 32, 245–251. [Google Scholar] [CrossRef] - Erdős, P.; Rényi, A. On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci.
**1960**, 5, 17–61. [Google Scholar] - Watts, D.J.; Strogatz, S.H. Collective dynamics of ‘small-world’networks. Nature
**1998**, 393, 440. [Google Scholar] [CrossRef] [PubMed] - Guimera, R.; Amaral, L.A.N. Functional cartography of complex metabolic networks. Nature
**2005**, 433, 895–900. [Google Scholar] [CrossRef] [PubMed] - Battiston, F.; Nicosia, V.; Latora, V. Structural measures for multiplex networks. Phys. Rev. E
**2014**, 89, 032804. [Google Scholar] [CrossRef] [PubMed] - Stella, M.; Brede, M. Mental Lexicon Growth Modelling Reveals the Multiplexity of the English Language. In Complex Networks VII; Springer: Berlin/Heidelberg, Germany, 2016; pp. 267–279. [Google Scholar]
- MacWhinney, B. The talkbank project. In Creating and Digitizing Language Corpora; Springer: Berlin/ Heidelberg, Germany, 2007; pp. 163–180. [Google Scholar]
- Hills, T.T.; Maouene, M.; Maouene, J.; Sheya, A.; Smith, L. Longitudinal analysis of early semantic networks: Preferential attachment or preferential acquisition? Psychol. Sci.
**2009**, 20, 729–739. [Google Scholar] [CrossRef] [PubMed] - Beckage, N.M.; Colunga, E. Language networks as models of cognition: Understanding cognition through language. In Towards a Theoretical Framework for Analyzing Complex Linguistic Networks; Springer: Berlin/Heidelberg, Germany, 2016; pp. 3–28. [Google Scholar]
- Beckage, N.; Smith, L.; Hills, T. Small worlds and semantic network growth in typical and late talkers. PLoS ONE
**2011**, 6, e19348. [Google Scholar] [CrossRef] [PubMed] - Sizemore, A.E.; Karuza, E.A.; Giusti, C.; Bassett, D.S. Knowledge gaps in the early growth of semantic networks. arXiv, 2017; arXiv:1709.00133. [Google Scholar]
- De Domenico, M.; Solé-Ribalta, A.; Omodei, E.; Gómez, S.; Arenas, A. Ranking in interconnected multilayer networks reveals versatile nodes. Nat. Commun.
**2015**, 6, 6868. [Google Scholar] [CrossRef] [PubMed] - Stephenson, K.; Zelen, M. Rethinking centrality: Methods and examples. Soc. Netw.
**1989**, 11, 1–37. [Google Scholar] [CrossRef] - Bozzo, E.; Franceschet, M. Resistance distance, closeness, and betweenness. Soc. Netw.
**2013**, 35, 460–469. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Mean closeness centrality and distance entropy of Erdos–Renyi (ER) random graphs of different node sizes and different link probabilities. Mean values are averaged for all nodes in a graph and across 100 different graph realisations. Link probabilities are relative to all node sizes. All different ensembles converge to the same pattern of distance entropy roughly above rewiring probability $p=0.6$. (

**b**) On a log–log scale, average shortest path length of ER random graphs for different link probabilities. The average distance decreases with increasing rewiring probability p and tipping points are evident around $p\approx 0.1$, after which the average shortest path length decreases with a slower rate.

**Figure 2.**Mean closeness centrality and distance entropy of Watts–Strogatz small-world (SW) networks of different node sizes and different rewiring probabilities. The rewiring probabilities plotted above are relative only to the case with $N=256$, and are provided only as a guideline. Mean values are averaged for all nodes in a graph and across 100 different independent realizations.

**Figure 3.**Mean closeness centrality and distance entropy of growing Barabasi–Albert (BA) networks for different values of the link growth rate m. Initial values are relative to networks with 100 nodes and are highlighted in red. Growing networks are measured once every 100 nodes are added. Simulated networks range from 100 up to 1500 nodes. Mean values are averaged for all nodes in a graph and across 100 independent realizations.

**Figure 4.**Distance entropy provides different centrality information on nodes, compared to closeness centrality. Here we consider a BA network with $N=25$ nodes and $m=4$. (

**a**) Nodes with low distance entropy are highlighted in green. (

**b**) Nodes with high closeness are highlighted in red. (

**c**) Cartography representing the distance entropy and closeness centrality of individual nodes in the network. Gray lines indicate quartiles. Nodes with the lowest (highest) distance entropy (closeness) are highlighted in green (red). The two sets of nodes do not overlap. Not considering distance entropy would lead to a closeness distribution reported in the top subpanel, where many nodes would end up displaying similar closeness centrality despite their different connectivity patterns, here highlighted by their distance entropy.

**Figure 5.**Distance entropy cartography for the $N=529$ words in the multiplex lexical network of young toddlers. Within a window of width w, nodes with similar closeness centrality can have quite different distance entropies.

**Table 1.**Improvements in word gains (relative to the reference closeness case) for different values of binning width w. p-values are relative to the observed improvement relative to a reference distribution. Reference distributions are obtained by ranking nodes at random (rather than through distance entropy). When $w>0.05$, no improvements are obtained.

Width w | Improvement (%) | p-Value |
---|---|---|

0 | 0 | 1 |

0.005 | +3.9% | 0.3 |

0.010 | +7.9% | 0.05 |

0.015 | +13.1% | 0.001 |

0.020 | +13.3% | 0.001 |

0.025 | +13.6% | 0.001 |

0.030 | +7.9% | 0.01 |

0.035 | +8.0% | 0.01 |

0.040 | +4.0% | 0.03 |

0.045 | +5.1% | 0.01 |

0.050 | +0.1% | 0.01 |

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Stella, M.; De Domenico, M.
Distance Entropy Cartography Characterises Centrality in Complex Networks. *Entropy* **2018**, *20*, 268.
https://doi.org/10.3390/e20040268

**AMA Style**

Stella M, De Domenico M.
Distance Entropy Cartography Characterises Centrality in Complex Networks. *Entropy*. 2018; 20(4):268.
https://doi.org/10.3390/e20040268

**Chicago/Turabian Style**

Stella, Massimo, and Manlio De Domenico.
2018. "Distance Entropy Cartography Characterises Centrality in Complex Networks" *Entropy* 20, no. 4: 268.
https://doi.org/10.3390/e20040268