*1.1. Literature Review*

Network theory is an important tool for describing and analyzing complex systems throughout the social, biological, physical, information and engineering sciences [1]. Originally, almost all studies of networks employed an abstraction in which systems are represented as ordinary graphs [2]. Although this approach is naive in many respects, it has been extremely successful. For instance, it has been used to illustrate that many real networks possess a heavy-tailed degree distribution [3], exhibit the small-world property [4], contain nodes that play central roles [5] and/or have modular structures [6].

It has been recently recognized [7–9] that most complex systems are not simply formed by a simple network, but they are instead formed by multilayer networks. Multilayer networks include not just one but several layers (networks) characterizing interactions of different nature and connotation. Multiplex networks [10,11] are a special type of multilayer networks. They are formed by a set of nodes connected by different types of interactions. The term multiplex was defined to indicate the presence of more than one relationship between the same actors of a social network [12]. Each set of interactions of the same type determines a distinct layer (network) of the multiplex. Examples of multiplex networks are ubiquitous. Other major examples of multiplex networks range from transportation networks [13] to social [14], financial [15] and biological networks [16]. In transportation networks, different layers can represent different means of transportation while in scientific collaboration networks the different layers can represent several topics of the collaboration. Given the surge of interest in multiplex networks, recently several algorithms [17,18] have been proposed to assess the centrality of nodes in these multilayer structures.

On the other hand, when pairs of nodes can be connected through multiple links and in multiple layers, the ranking of nodes should necessarily reflect the importance of nodes in one layer as well as their importance in other interdependent layers. Our world is increasingly dependent on efficient ranking algorithms [19–21]. Currently the ranking of nodes in complex networks is used in a variety of different contexts [22], from finance to social, urban and biological networks. For instance, in the context of economical trade networks, formed by networks of countries and products, ranking algorithms [23] are recognized as an important tool to evaluate the economic development of countries.

Within the general problem of centrality in urban networks [24], the adapted PageRank algorithm (APA) proposed by [25] provides us a model to establish a ranking of nodes in spatial networks according to their importance in it. This centrality was originally proposed for urban networks, although it may be generalized to spatial networks or networks with data. It constitutes a centrality measure in urban networks with the main characteristic that it is able to consider the importance of data obtained from any source in the whole process of computing the centrality of the individual nodes. Starting from the basic idea of the PageRank vector concept [26], the matrix used for obtaining the classification of the nodes is constructed in a slightly different way as we will see later.

Centrality measures originally defined on single networks have been used extensively in some types of networks, as social, technological or biological. In multiplex networks these measures can be extended in different ways [27]. Generalizing centrality measures from monoplex networks to multilayer networks is not trivial. When ranking nodes in a multilayer or multiplex network, the key question to be addressed is how one should take into consideration all the different types of edges, not all of which have the same importance [17].

The Multiplex PageRank [28] evaluates the centrality of the nodes of multiplex networks and is based on the idea of biased random walks to define the Multiplex PageRank centrality measure in which the effects of the interplay between networks on the centrality of nodes are directly taken into account. Other centrality measures associate a different influence with the links of different layers that weights their contribution to the centrality of the nodes, as for example Multiplex Eigenvector Centralities [17], or Functional Multiplex PageRank [18], which takes into account the fact that different multilinks contribute differently to the centrality of each node and associates with each node a function. In [29], Solé–Ribalta et al. re-define the betweenness centrality measure to account for the inherent structure of multiplex networks and propose an algorithm to calculate it efficiently.

### *1.2. Main Contribution*

The main objective of this paper is to propose a centrality measure for biplex networks that adapts the APA centrality for spatial networks with data to the PageRank two-layer approach.

The multiplex PageRank algorithm proposed by Halu et al. [28] measures the centrality of a node in a layer *β* with the corresponding adjacency matrix using a random walker as in the usual PageRank biased by the PageRank of the nodes in the layer *alpha*. So, Multiplex PageRank can be described in terms of the bias that one layer exerts on the random jumps that a surfer makes in another layer. Four versions of the centrality are presented depending on how layers affect each other or, alternatively, exert a bias upon the random walk.

The advantage of the model proposed for biplex networks is that since it is based on the APA centrality, the randomness in the jumps of the surfer is replaced by the influence of the data present in the network. Therefore, the model is able to study and analyze several relationships of a set of nodes by different layers, but it is also capable of measuring the influence of the data in the different layers of the network. Thus, it is not only taken into account the topology of the networks but also the data. In addition, the *α* parameter initially associated to the PageRank model and exported to APA centrality can be used to assign the importance that each of these data layers has in the calculation of centrality. These characteristics summarizes the novelty and potential of this research work.

The potential of the applications to which this centrality can be applied should also be highlighted. Nowadays multilayers and multiplex networks are investigated in many fields, as were enumerated in Section 1.1. However, there are other potential applications not so well known but equally interesting where the centrality measure proposed in this research can be applied. One of these topics is the improvement in the recommender systems [30,31]. These systems were initially based on demographic, content-based and collaborative filtering although actually are incorporating social information. Another unknown topic of application is in Public Safety Networks [32,33] which have emerged as the key solution to a successful response to emergency and disaster events. In these systems each user is associated with some social, physical and mobility-related characteristics and attributes in a public safety network.

### *1.3. Structure of the Paper*

This paper is organized as follows: Section 2 summarizes the methodology used in the paper: the basic characteristics of the APA centrality, as well as the biplex approach to the PageRank vector. Once these models are described, the new centrality is presented combining the APA centrality (based on data) and the biplex approach of PageRank vector. In Section 3, some results are presented with their interpretation. The discussion of the experimental results are performed in Section 4 and, finally, some conclusions are summarized.
