Bayrakdar Nedim, Gemmetto Valerio, Garlaschelli Diego
Lorentz Institute for Theoretical Physics, University of Leiden, 2333 CA Leiden, The Netherlands.
IMT School of Advanced Studies Lucca, 55100 Lucca, Italy.
Entropy (Basel). 2023 May 22;25(5):828. doi: 10.3390/e25050828.
Multilayer networks represent multiple types of connections between the same set of nodes. Clearly, a multilayer description of a system adds value only if the multiplex does not merely consist of independent layers. In real-world multiplexes, it is expected that the observed inter-layer overlap may result partly from spurious correlations arising from the heterogeneity of nodes, and partly from true inter-layer dependencies. It is therefore important to consider rigorous ways to disentangle these two effects. In this paper, we introduce an unbiased maximum entropy model of multiplexes with controllable intra-layer node degrees and controllable inter-layer overlap. The model can be mapped to a generalized Ising model, where the combination of node heterogeneity and inter-layer coupling leads to the possibility of local phase transitions. In particular, we find that node heterogeneity favors the splitting of critical points characterizing different pairs of nodes, leading to link-specific phase transitions that may, in turn, increase the overlap. By quantifying how the overlap can be increased by increasing either the intra-layer node heterogeneity (spurious correlation) or the strength of the inter-layer coupling (true correlation), the model allows us to disentangle the two effects. As an application, we show that the empirical overlap observed in the International Trade Multiplex genuinely requires a nonzero inter-layer coupling in its modeling, as it is not merely a spurious result of the correlation between node degrees across different layers.
多层网络表示同一组节点之间的多种连接类型。显然，只有当多重网络不仅仅由独立层组成时，对系统的多层描述才会增加价值。在现实世界的多重网络中，预计观察到的层间重叠可能部分源于节点异质性产生的虚假相关性，部分源于真正的层间依赖性。因此，考虑用严格的方法来区分这两种效应很重要。在本文中，我们引入了一种具有可控层内节点度和可控层间重叠的多重网络无偏最大熵模型。该模型可以映射到一个广义伊辛模型，其中节点异质性和层间耦合的结合导致了局部相变的可能性。特别是，我们发现节点异质性有利于表征不同节点对的临界点的分裂，导致特定于链接的相变，进而可能增加重叠。通过量化增加层内节点异质性（虚假相关性）或层间耦合强度（真实相关性）如何增加重叠，该模型使我们能够区分这两种效应。作为一个应用，我们表明，在国际贸易多重网络中观察到的经验重叠在其建模中确实需要非零的层间耦合，因为它不仅仅是不同层节点度之间相关性的虚假结果。
