Complex network is an interdiscipline that describes the complex relationship in our real world. A large number of empirical studies indicate that many real-world systems can be abstracted as complex networks. They recover the rich diversity of systems and are found to have lots of common structural features.

A random model of networks with specific degree sequences is often considered as a zero model to test the structural properties of complex networks. In order to capture the inner structures and behaviors of real-world network systems, along with the degree distribution, the mixing patterns in networks are also essential.

It plays a vital role in many fields, such as mean distance, robustness, stability, percolation thresholds, epidemic spreading and synchronization of oscillators.

By studying the statistical parameters of the two ends of edges in a network, such as the correlation coefficient and the mean value, we can obtain the joint degree probability distribution, which is decisive for the structure, function and dynamics of a network.

To measure the mixing patterns in complex networks, Newman introduced a method based on the Pearson correlation coefficient. However, further studies indicate that the Pearson coefficient has a serious drawback. Its value critically depends on the size and degree distribution of networks. In particular, it would converge to zero for large scale-free networks. This drawback strongly impedes the quantitative comparison of different networks. This point cannot be ignored, as the size of modern networks is getting larger and larger, for instance, scientific collaboration networks and the World-Wide Web.

In addition, when given the degree sequence and the correlation coefficient, the traditional way to get such a network is the rewiring procedure, but it is inefficiency. Therefore, searching for methods which can generate networks with priori correlation coefficients is very meaningful to the research of complex networks.

In this thesis, to solve the above drawback of the Pearson-coefficient-based measure, we first analyzed the value range of the correlation coefficient of any α-order rα.

We analytically obtained the range of rα for scale-free networks with power-law exponent γ>2 in the large network limit N→∞. We obtain that if and only if α<γ-2/2, both the lower bound rα, min and the upper bound rα, max do not approach zero. Further, if and only if α=2-γ, which stands for the Spearman coefficient, both bounds achieve their maximum value -1 and 1 respectively. In the real physical world, the data we are exposed to on a daily basis is generally continuous and analytical. Expanded the indexes of a system with Taylor series, the degree correlation coefficient corresponds to the second-order derivation. Examining its value range and validity could help us with understanding other indicators.

For complex network researches, the random network model is the most basic network model. For further research, generating a network with any prescribed coefficient is crucial. In order to get networks with different assortativity coefficients, one of the methods is reshuffling the edges in a network. We first studied the reshuffling networks and obtained a unified expression for the joint probability distribution of different assortativity reshuffling probability. Results show that it is doubtful whether networks obtained by reshuffling are uniform. In other words, the reshuffling method is not the most intuitive one in terms of relevance at least. Furthermore, we propose a novel method based on Spearman rank correlation coefficient to measure the mixing patterns in complex networks.

In mathematics, Spearman coefficient is usually used to quantify how well two columns of data monotonically depend on each other. It is rank-based, nonparametric, independent of the network size and degree distribution.

Hence it could overcome the aforementioned drawback of Pearson coefficient and work better. Moreover, we discover that the normalized rank orders of stubs are statistically in a linear correlation, and the correlation coefficient is just the Spearman coefficient. We argue that this kind of correlation universally exists in complex networks, and we verify it on both empirical and artificial networks. It can be used as an essential factor to determine the point degree probability distribution of networks. Based on the linear relationship above and Marrows' results on ranking models, we give a simple exponent and Gauss approximate form to describe the jointed degree probability distribution of networks. With the jointed degree probability distribution, we could directly generate a network with any prescribed Spearman coefficient. We test the exponent and the Gauss model with simulations, and find them in good agreement with the theory. Moreover, comparing with the classical rewiring procedure, networks created by the exponent and Gauss model also have good randomness and the computation complexity is reduced.

Finally, we analyze national trade network data. We combine the proximity indicator and the fitness-complexity algorithm, to study what industries can possibly improve a country's fitness. We use the “proximity” metric to find the products that a country is capable of developing. Further, we define a list of “core products” which not only have high export volumes, but also are complex products which have relatively high complexity. These “core products” are regarded as the target products to enhance the country's fitness. And we attempt to recommend the relevant products for the developing countries who have a higher probability to produce according to the proximity of the product space.

When? | 10.05.2023 14:15 |
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Where? | PER 08 1.50 Chemin du Musée 3, 1700 Fribourg |

speaker | Wenyao Zhang, Présentation publique de thèse de doctorat
Groupe Zhang |

Contact | Département de physique, groupe Zhang Prof. Zhang yi-cheng.zhang@unifr.ch |