Weighted multiplex networks

Abstract

One of the most important challenges in network science is to quantify the information encoded in complex network structures. Disentangling randomness from organizational principles is even more demanding when networks have a multiplex nature. Multiplex networks are multilayer systems of [Formula: see text] nodes that can be linked in multiple interacting and co-evolving layers. In these networks, relevant information might not be captured if the single layers were analyzed separately. Here we demonstrate that such partial analysis of layers fails to capture significant correlations between weights and topology of complex multiplex networks. To this end, we study two weighted multiplex co-authorship and citation networks involving the authors included in the American Physical Society. We show that in these networks weights are strongly correlated with multiplex structure, and provide empirical evidence in favor of the advantage of studying weighted measures of multiplex networks, such as multistrength and the inverse multiparticipation ratio. Finally, we introduce a theoretical framework based on the entropy of multiplex ensembles to quantify the information stored in multiplex networks that would remain undetected if the single layers were analyzed in isolation.

Figure 1. Example of all possible multilinks in a multiplex network with layers and nodes.

Nodes and are linked by one multilink .

Figure 2. Average multistrength and average inverse multiparticipation ratio versus multidegree in the CoCo-PRE/PRL multiplex network.

The average multistrengths and the average inverse multiparticipation ratios are fitted by a power-law distribution of the type described in Eq. (8) (fitted distributions are here indicated by black dashed lines). Statistical tests for the collaboration network of PRL suggest that the exponents defined in Eq. (8) are the same, while exponents are significantly different. Similar results can be obtained for the exponents in the PRE collaboration layer. Nevertheless, multistrengths are always larger than multistrengths and , when multistrengths are calculated over the same number of multilinks, i.e., (see Text S1 for the statistical test on this hypothesis).

Figure 3. Properties of multilinks in the weighted CoCi-PRE multiplex network.

In the case of the collaboration network, the distributions of multistrengths versus multidegrees always have the same exponent, but the average weight of multilinks is larger than the average weight of multilinks . Moreover, the exponents , are larger than exponents . In the case of the citation layer, both the incoming multistrengths and the outgoing multistrengths have a functional behavior that varies depending on the type of multilink. Conversely, the average inverse multiparticipation ratio in the citation layer does not show any significant change of behavior when compared across different multilinks.

Figure 4. (A) Value of the indicator defined in Eq.

(10) indicating the amount of information carried by the correlated and the uncorrelated multiplex ensembles of nodes with respect to a null model in which the weights are distributed uniformly over the multiplex network. (B) Value of the indicator defined in Eq. (12) indicating the additional amount of information encoded in the properties of multilinks in the correlated multiplex ensemble with respect to the corresponding uncorrelated multiplex ensemble. The solid line refers to the average value of over the different multiplex network sizes.