Gemmetto V, Garlaschelli D
Instituut-Lorentz for Theoretical Physics, Leiden Institute of Physics, University of Leiden, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands.
Sci Rep. 2015 Mar 13;5:9120. doi: 10.1038/srep09120.
Several systems can be represented as multiplex networks, i.e. in terms of a superposition of various graphs, each related to a different mode of connection between nodes. Hence, the definition of proper mathematical quantities aiming at capturing the added level of complexity of those systems is required. Various steps in this direction have been made. In the simplest case, dependencies between layers are measured via correlation-based metrics, a procedure that we show to be equivalent to the use of completely homogeneous benchmarks specifying only global constraints. However, this approach does not take into account the heterogeneity in the degree and strength distributions, which is instead a fundamental feature of real-world multiplexes. In this work, we compare the observed dependencies between layers with the expected values obtained from maximum-entropy reference models that appropriately control for the observed heterogeneity in the degree and strength distributions. This information-theoretic approach results in the introduction of novel and improved multiplexity measures that we test on different datasets, i.e. the International Trade Network and the European Airport Network. Our findings confirm that the use of homogeneous benchmarks can lead to misleading results, and highlight the important role played by the distribution of hubs across layers.
若干系统可表示为多重网络，即由各种图叠加而成，每个图对应节点间不同的连接模式。因此，需要定义合适的数学量来捕捉这些系统增加的复杂程度。在这个方向上已经取得了各种进展。在最简单的情况下，层间依赖关系通过基于相关性的指标来衡量，我们证明这个过程等同于使用仅指定全局约束的完全同质基准。然而，这种方法没有考虑度分布和强度分布的异质性，而这却是现实世界多重网络的一个基本特征。在这项工作中，我们将观察到的层间依赖关系与从最大熵参考模型获得的期望值进行比较，这些模型适当地控制了观察到的度分布和强度分布的异质性。这种信息论方法导致引入了新的和改进的多重性度量，我们在不同的数据集上进行了测试，即国际贸易网络和欧洲机场网络。我们的研究结果证实，使用同质基准可能会导致误导性结果，并突出了枢纽在各层间分布所起的重要作用。
