Functional Multiplex PageRank

Recently it has been recognized that many complex social, technological and biological networks have a multilayer nature and can be described by multiplex networks. Multiplex networks are formed by a set of nodes connected by links having different connotations forming the different layers of the multiplex. Characterizing the centrality of the nodes in a multiplex network is a challenging task since the centrality of the node naturally depends on the importance associated to links of a certain type. Here we propose to assign to each node of a multiplex network a centrality called Functional Multiplex PageRank that is a function of the weights given to every different pattern of connections (multilinks) existent in the multiplex network between any two nodes. Since multilinks distinguish all the possible ways in which the links in different layers can overlap, the Functional Multiplex PageRank can describe important non-linear effects when large relevance or small relevance is assigned to multilinks with overlap. Here we apply the Functional Page Rank to the multiplex airport networks, to the neuronal network of the nematode C. elegans, and to social collaboration and citation networks between scientists. This analysis reveals important differences existing between the most central nodes of these networks, and the correlations between their so called pattern to success. Introduction. – Many complex interacting systems are formed by nodes related by different types of interactions forming multiplex networks [1–4]. Examples of multiplex networks are ubiquitous, from social [5–8] to transportation [9–11] and biological networks [9,12]. For example scientific authors form at the same time collaboration networks and citation networks in which they cite each other [7,8], the airport network is formed by airports connected by flights operated by different airline companies [10], in the brain neurons are simultaneously connected by chemical and electrical types of connections [9, 13, 14]. A multiplex network is therefore constituted by a set of N nodes interacting through M layers which are networks formed by links having the same connotation. In recent years we have gained a significant understanding of the interplay between the structure and the dynamics of multiplex networks [15–21] and relevant insights regarding the level of information encoded in their correlated structure [5, 7–9, 22–24]. In this context, given the increasing number of multiplex network datasets, establishing the centrality of the nodes in multiplex networks has become a problem of major interest. Until now, several multiplex centrality measures have been proposed [8, 25–30] which aim at going beyond the definition of centrality in single networks [31–33]. Examples of multiplex centrality measure include the Versatility of the nodes [25], the Multiplex PageRank [8, 26], and the Eigenvector multiplex centrality [27]. The Versatility [25] emphasizes the relevance of nodes connected in many different layers and it applies to multiplex networks where corresponding nodes in different layers are connected by interlinks. The Multiplex PageRank [8, 26] exploits the correlations existing between the degree of the nodes in different layers through the use of a biased random walk. The Eigenvector multiplex centrality [27] instead assumes that the centrality of a node with respect to one layer is influenced by its centrality in other layers weighted by a matrix of influences that one layer has on the other layer. Both the Multiplex PageRank and the Eigenvector multiplex centrality do not make explicit use of the interlinks. The main challenge when defining a centrality of the nodes in a multiplex network without interlinks is that the centrality depends on the relevance