Abstract We formulate and propose an algorithm (MultiRank) for the weighted ranking of nodes and layers in large multiplex networks. MultiRank takes into account the full multiplex network structure of the data and exploits the dual nature of the network in terms of nodes and layers. The proposed centrality of the layers (influences) and the centrality of the nodes are determined by a coupled set of equations. The basic idea consist on assigning more centrality to nodes that are linked to central nodes in highly influential layers. The layers are more influential if highly central nodes are active in them. The algorithm applies to directed/undirected as well as to weighted/unweighted multiplex networks. We discuss the application of MultiRank to three major examples of multiplex network data sets: the European Air Transportation Multiplex Network, the Pierre Auger Multiplex Collaboration Network and the FAO Multiplex Trade Network. 1. Introduction Our world is increasingly dependent on efficient ranking algorithms [1–6]. Currently ranking algorithms based on PageRank [1] constitute by far the most diffuse way to navigate through online-stored information. As such they are perhaps one of the most significant revolutions in the field of knowledge exploration since the first publication of the Diderot Encyclopaedia. Currently the ranking of nodes in complex networks is used in a variety of different contexts [3–6], from finance to social and biological networks. In the context of economical trade networks, formed by bipartite networks of countries and products, ranking algorithms [7–9] are recognized as an important tool to evaluate the economic development of countries. Interestingly, the algorithms proposed for ranking nodes and products in bipartite trade networks have been shown to be also quite useful for other applications such as ranking species in bipartite ecological networks [10]. However it has been recently recognized [11–17] that most complex systems are not simply formed by a simple or a bipartite 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 [18–26] are a special type of multilayer networks. They are formed by a set of $$N$$ nodes connected by $$M$$ different types of interactions. Each set of interactions of the same type determines a distinct layer (network) of the multiplex network. Examples of multiplex networks are ubiquitous. For instance trade networks [27–30] between countries are inherently multiplex structures formed by $$N$$ nodes (countries) trading with each other $$M$$ different kinds of products (layers). Other major examples of multiplex networks range from transportation networks [24, 31] to social [20, 22, 32–34], financial [35, 36] and biological networks [16, 30, 37, 38]. In transportation networks [24, 31], different layers can represent different means of transportation while in scientific collaboration networks the different layers can represent different topics of the collaboration [22, 34]. Finally, in biology, multiplex networks span from the multiplex connectome of Caenorhabditis elegans to multiplex molecular networks in the cell [16, 30, 37–39]. Given the surge of interest in multiplex networks, recently several algorithms [40–46] have been proposed to assess the centrality of nodes in these multilayer structures. In Ref. [40, 41], the proposed algorithm captures how the centrality of the nodes in a given layer (master layer) of the multiplex network can affect the centrality of the nodes in other layers. This effect is modelled by considering a PageRank algorithm biased on the centrality of the nodes in the master layer. In Ref. [42], it has been proposed instead to rank simultaneously nodes and layers of the multiplex network based on a random walk process [23, 24] (defined over a 2-covariant and 2-contravariant tensor representation of the multiplex network [17]) that hops between nodes of the same layer and between nodes of different layers as well. The resulting centrality called ‘versatility’ strongly awards nodes active (connected) in many layers. Moreover in Ref. [45], the communicability has been proposed as a centrality measure based on the flow of information in multiplex networks. Here a different approach is considered, where we consider a random walk hopping through links of different layers with different probabilities determined by the centrality of the layers (influences). The role of the influences of the layers has been first introduced in Ref. [43] where different measures for the centrality of the nodes given a set of influences of the layers have been proposed. Knowing the correct values to attribute to the influences of the layers might actually be a non-trivial task as in most of the cases the influences are not known a priori. In order to address this issue in Ref. [44], we proposed the Functional Multiplex PageRank that allows to take a comprehensive approach and to establish the centrality of the nodes for a wide range of influence values. Specifically, the Functional Multiplex PageRank attributes not only a single number (rank) to a node but an entire function determining the rank of the node as a function of the influences attributed to its different types of links. Although the Functional Multiplex PageRank [44] has been shown to be very informative for multiplex networks with few layers, its applications to multiplex networks with many layers is somewhat limited. Here in order to address this question we propose a ranking algorithm, MultiRank, that is specified by a coupled set of equations that simultaneously determine the centrality of the nodes and the influences of the layers of a multiplex network. The MultiRank algorithm applies to any type of multiplex network including weighted and directed multiplex network structures. Interestingly the MultiRank can be related to the literature of ranking algorithms for bipartite networks [7–9]. In fact, while processing all the information present in the multiplex network structure, it also exploits the bipartite nature [21] of the network between nodes and layers that can be extracted directly from the multiplex. This bipartite network indicates which node are connected in which layer and which is the strength of their incoming and outgoing links. The MultiRank algorithm is based on a random walk among the nodes of the multiplex network. Starting from a node the random walker either hops to a neighbour node with probability depending on the influence of the layer or jumps to a random node of the network that is not isolated. The influences are the layer centralities that are determined by another set of equations, depending on the centrality of the nodes that are active (i.e. connected) in a given layer. MultiRank efficiently ranks nodes and layers of large multiplex networks including many layers. Here, we discuss the application of the algorithm to three major examples of multiplex networks: the European Air Transportation Multiplex Network [31], the Pierre Auger Multiplex Collaboration Network [34] and the Food and Agriculture Organization (FAO) Multiplex Trade Network [30]. 2. The MultiRank algorithm 2.1 MultiRank: the motivation Let us consider a multiplex network of $$M$$ layers $$\alpha=1,2,\ldots, M$$ and $$N$$ nodes $$i=1,2,3,\ldots, N$$. We indicate with $${\bf A}^\left[{\alpha}\right]$$ the adjacency matrix (eventually directed and weighted) of layer $$\alpha$$ with elements $$A_{ij}^\left[{\alpha}\right]>0$$ if node $$i$$ points to node $$j$$ with weight $$A_{ij}>0$$. The aggregated network is the single network in which each pair of nodes is connected if they have at least a link in one layer of the multiplex network. Different layers might have more or less connections and/or more or less total weight of their links. Let us characterize the potential heterogeneity of the layers by attributing to each layer $$\alpha$$ the quantity $$W^\left[{\alpha}\right]$$ defined as \begin{align} W^\left[{\alpha}\right]=\sum_{i=1}^N \sum_{j=1}^N A_{ij}^\left[{\alpha}\right]. \end{align} (1) For an undirected unweighted multiplex network $$W^\left[{\alpha}\right]$$ indicates the double of the total number of links in layer $$\alpha$$. For a directed unweighted multiplex network it indicates the total number of links in layer $$\alpha$$. For a weighted multiplex network it either indicates the total weight of the links in layer $$\alpha$$ (directed case) or the double of this number (undirected case). Our aim is to introduce a framework to determine simultaneously the centrality $$X_i$$ of the nodes $$i=1,2,3,\ldots, N$$ and the centrality $$\bf z^\left[{\alpha}\right]$$ of the layers (influences) $$\alpha=1,2,\ldots M$$. On one side, we will interpret the multiplex network as a coloured network where different types of links are assigned a different influence. On the other side we will exploit the weighted, directed bipartite network formed by nodes and layers [11, 21] that can be constructed from the multiplex network. This bipartite network will provide us with information about the activity of the nodes [22] in each layer, that is if they are present (and therefore connected) in the layer. Additionally, the weighted and directed links of this bipartite network will indicate which is the in-strength and out-strength of each node in any given layer. In Fig. 1, we display graphically how the bipartite network between nodes and layers can be constructed starting from any multiplex network data. Fig. 1. View largeDownload slide The schematic representation of a multiplex network formed by three different layers is here shown together with the construction of the directed bipartite network formed by nodes and layers and the multiplex representation as a coloured network. The directed bipartite network indicates for each node in which layers the node is connected. Additionally, as it is explained in the text it gives direct information about in-strengths and out-strength of each node in any given layer. Fig. 1. View largeDownload slide The schematic representation of a multiplex network formed by three different layers is here shown together with the construction of the directed bipartite network formed by nodes and layers and the multiplex representation as a coloured network. The directed bipartite network indicates for each node in which layers the node is connected. Additionally, as it is explained in the text it gives direct information about in-strengths and out-strength of each node in any given layer. Specifically, we will work with two matrices extracted from the multiplex network. The first matrix is the $$N\times N$$ weighted matrix $${\bf G}$$ of the coloured network, where the links of each layer $$\alpha=1,2,\ldots, M$$ are weighted with the influences $$z^\left[{\alpha}\right]$$ associated to it. Therefore the elements $$G_{ij}$$ of the matrix $${\bf G}$$ are given by \begin{align} G_{ij}=&\sum_{\alpha=1}^MA_{ij}^\left[{\alpha}\right]z^\left[{\alpha}\right]. \label{B1} \end{align} (2) Secondly, we extract the incidence matrices of the bipartite network constructed from the multiplex network by considering the connectivity of each node $$i$$ in layer $$\alpha$$. For directed multiplex networks we need to distinguish two $$M\times N$$ incidence matrices $${\bf B}^{\rm in}$$ and $${\bf B}^{\rm out}$$ of elements \begin{align} B_{\alpha i}^{\rm in}=&\frac{\sum_{j}A_{ji}^\left[{\alpha}\right]}{W^\left[{\alpha}\right]},\nonumber\\ B_{\alpha i}^{\rm out}=&\frac{\sum_{j}A_{ij}^\left[{\alpha}\right]}{W^\left[{\alpha}\right]}. \end{align} (3) Therefore $$B_{\alpha i}^{\rm in}$$ indicates the normalized in-strength of node $$i$$ in layer $$\alpha$$ and $$B_{\alpha i}^{\rm out}$$ indicates the normalized out-strength of node $$i$$ in layer $$\alpha$$. For undirected multiplex networks the matrices $${\bf B}^{\rm in}$$ and $${\bf B}^{\rm out}$$ are identical. Note that if node $$i$$ is not connected in layer $$\alpha$$ we have $$B_{\alpha i}^{\rm in}=B_{\alpha i}^{\rm out}=0,$$ and we say that the node is inactive in layer $$\alpha$$ [21, 22]. Otherwise, if node $$i$$ is connected in layer $$\alpha$$, we will say that the node is active in layer $$\alpha$$. We assume that the centrality of a node depends on the centrality of the layers in which it is connected, and additionally we assume that the centrality of the layers depends on the centrality of the nodes that are active on it. Specifically the centrality of a given node increases when already central nodes point to it in layers with high influence. Therefore, given the influences $$z^\left[{\alpha}\right]$$ of the layers $$\alpha$$ the centrality $$X_i$$ of a node can be taken for instance to be determined by the PageRank centrality associated to the weighted adjacency matrix $${\bf G}$$. In this case, the centralities of the nodes are determined by the steady state of a random walker that, starting from a node $$j$$, with probability $$\tilde{\alpha}$$ hops to a neighbour node $$i$$ of the aggregated network chosen with probability proportional to $$G_{ji}$$ and with probability $$1-\tilde{\alpha}$$ jumps to a random node which is not isolated in the weighted network described by the adjacency matrix $${\bf G}$$. Therefore the equations for the centrality $$X_i$$ of the node $$i$$ given the influences $$z^\left[{\alpha}\right]$$ of the layers read \begin{align} X_i=&\tilde{\alpha} \sum_{j=1}^N \frac{G_{ji}}{\kappa_j} X_j+\beta v_i, \label{PR} \end{align} (4) where $$\tilde{\alpha}$$ is taken to be $$\tilde{\alpha}=0.85$$ (as usual in the context of the PageRank algorithms) and $$\kappa_j$$, $$v_i$$ and $$\beta$$ are given by \begin{align} \kappa_j = &\max\left(1,\sum_{i=1}^N G_{ji}\right)\!,\nonumber\\ v_i = &\theta\left(\sum_{j=1}^N [G_{ij}+G_{ji}]\right)\!,\nonumber\\ \beta = &\frac{1}{\sum_{i=1}^N v_i}\sum_{j=1}^N \left[1-\tilde{\alpha}\ \theta\left(\sum_{i=1}^N G_{ji} \right)\right]X_j. \label{defc} \end{align} (5) This algorithm is a variation of the recently formulated Functional Multiplex PageRank [44] as long as the influences of the layers $$\{z^\left[{\alpha}\right]\}$$ are considered as free parameters. (Note that also in [44] the influences are not simply attributed to layers but they are assignment to each different pattern of connection that can exist between two nodes of the multiplex, i.e. every type of possible multilink.) In particular, the Functional Multiplex PageRank determines the centrality of each node as a function of the set of influences associated to the layers of the multiplex network. This approach, although quite useful in the context of multiplex networks with few layers, can become prohibitive in the case of multiplex networks with many layers, that is for $$M\gg 1$$. Therefore, here we propose to couple the Eqs. (4) for the centrality $$X_i$$ of a generic node $$i$$ of the multiplex network with another set of equations determining the best value for the influences of the layers. In order to determine these equations, we will exploit the nature of the bipartite network between nodes and layers. Our starting point will be that layers are more influential if highly central nodes are active in them. The simplest equation enforcing this principle is therefore \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}W^\left[{\alpha}\right]\sum_{i=1}^N B_{\alpha i}^{\rm in}X_i \label{linear} \end{align} (6) with $${\mathcal N}$$ indicating a normalization constant. First of all, we note here that this equation for $$z^\left[{\alpha}\right]$$ rewards layers with larger total weight $$W^\left[{\alpha}\right]$$. For unweighted multiplex networks this implies that layers with more links are more influential while for weighted multiplex networks this implies that the layers with more total weight are more influential. Additionally, this equation establishes layers having a larger number of central nodes with large in-strength as more influential. We note here that this algorithm is inspired by the algorithm proposed in Ref. [7] for ranking countries and products in bipartite trade networks but differs most notably from that model because the Eq. (4) for the centrality of the nodes $$X_i$$ depends on the full underlying multiplex network structure, that is it depends not only on the layers where node $$i$$ is active but also on the connections between different nodes existing in those layers. The linear equation (6) for the centrality of the nodes $$X_i$$ can be modified by including a tuning parameter $$\gamma>0$$ as in the following \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}W^\left[{\alpha}\right]\sum_{i=1}^N B_{\alpha i}^{\rm in}[X_i]^{\gamma}, \label{gamma} \end{align} (7) where $${\mathcal N}$$ indicates a normalization constant. For $$\gamma=1$$ (linear case) we recover Eq. (6). For $$\gamma<1,$$ the nodes with low centrality contribute more than in the linear case while for $$\gamma>1$$ their contribution is suppressed with respect to the linear case. Equation (7) can be coupled to Eq. (4) to get a simultaneous ranking of nodes $$\{X_i\}$$ and layers $$\{z^\left[{\alpha}\right]\}$$. Specifically the algorithm can be used in the following cases: (i) layers with larger total weight $$W^\left[{\alpha}\right]$$ are assumed to be more influential; (ii) layers with more active nodes of high centrality and high in-strength are considered more influential. However, sometimes we might desire to assign to the layers of a multiplex network an influence that evaluates the centrality of the layers independently of their different density and/or total weight, that is independently of $$W^\left[{\alpha}\right]$$. In this case, we propose the following normalized equations \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}\sum_{i=1}^N B_{\alpha i}^{\rm in}[X_i]^{\gamma} \label{gamma2} \end{align} (8) with $${\mathcal N}$$ indicating a normalization constant. Therefore, Eq. (8) can be used when the ranking of the layers needs to satisfy the following two conditions: (i) the influence of each layer is required to be independent of $$W^\left[{\alpha}\right]$$; (ii) layers with more active nodes of high centrality and high in-strength are considered more influential. A different approach consists in attributing more influence to elite layers with few highly central nodes with incoming links. In the formulation, of these alternative equations for the determination of the influences $$\{z^\left[{\alpha}\right]\}$$ of the layers, we have been inspired by the recently proposed algorithm [8, 9] for ranking nodes in the bipartite network of countries and products. Specifically, it is possible to propose the following set of equations for determining the values of the layers influences, \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}W^\left[{\alpha}\right]\left[\sum_{i=1}^N B_{\alpha i}^{\rm in}\left(X_i\right)^{-1}\right]^{-1}, \label{gamma30} \end{align} (9) where $${\mathcal N}$$ is a normalization constant. Here layers with more overall weights $$W^\left[{\alpha}\right]$$ are more influential, and whereas the overall weight of the layers is the same, layers with fewer highly central nodes receiving in-links are more influential. For instance in the Trade Multiplex Network a product is more important if the overall world-wide trade is very conspicuous for that product and only few central nodes are exporting it. Similarly to the previously discussed case, we can also introduce a parameter $$\gamma>0$$ and consider the following equations for the influences $$z^\left[{\alpha}\right]$$ of the layers, \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}W^\left[{\alpha}\right]\left[\sum_{i=1}^N B_{\alpha i}^{\rm in}\left(X_i\right)^{-\gamma}\right]^{-1}, \label{gamma3} \end{align} (10) where $${\mathcal N}$$ is a normalization constant. For $$\gamma=1$$ (linear case) we recover Eq. (9). For $$\gamma\neq 1,$$ the contribution of low centrality nodes in determining the centrality of the layers in which they are active can be either enhanced ($$\gamma>1$$) or suppressed $$(\gamma<1)$$. Equations (10) can be used if the ranking of the layer must satisfy the following set of conditions: (i) layers with larger total weight $$W^\left[{\alpha}\right]$$ are associated to a larger influence; (ii) layers with fewer active nodes of high centrality are considered more influential. Alternatively, we might consider equations in which we normalize by the total weights of the layers $$W^\left[{\alpha}\right]$$, getting \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}\left[\sum_{i=1}^N B_{\alpha i}^{\rm in}\left(X_i\right)^{-\gamma}\right]^{-1}. \label{gamma4} \end{align} (11) This last set of equations can be used when the ranking of the layers must satisfy the following conditions: (i) the influence of each layer is required to be independent of $$W^\left[{\alpha}\right]$$. (ii) layers with more active nodes of high centrality and high in-strength are considered more influential. 2.2 MultiRank: the definition Summarizing the discussion of the previous section here we introduce the MultiRank algorithm depending on three parameters: $$\gamma>0$$ , $$s$$ taking values $$s=1, -1$$ and $$a$$ taking values $$a=1,0$$. The MultiRank algorithm assigns a centrality $$X_i$$ to each node $$i$$ and an influence $$z^\left[{\alpha}\right]$$ to each layer $$\alpha$$ which can be found by solving the following set of coupled equations \begin{align} X_i=&\tilde{\alpha} \sum_{j=1}^N \frac{G_{ji}}{\kappa_j} X_j+\beta v_i\label{PR2} \end{align} (12) \begin{align} z^\left[{\alpha}\right]=&\frac{1}{\mathcal N}\left[W^\left[{\alpha}\right]\right]^{a}\left[\sum_{i=1}^N B_{\alpha i}^{\rm in}\left(X_i\right)^{s\gamma}\right]^{s} \label{z} \end{align} (13) where $$\tilde{\alpha}$$ is taken to be $$\tilde{\alpha}=0.85$$ and $$\kappa_j$$, $$v_i$$ and $$\beta$$ are given by Eq. (5) and $${\mathcal N}$$ is a normalization constant. For $$a=1$$ the influence of a layer is rewards layers with larger $$W^\left[{\alpha}\right]$$. For $$a=0$$, the influence of a layer is normalized with respect to $$W^\left[{\alpha}\right]$$. For $$s=1$$ layers have larger influence if they include more central nodes. In this case, the parameter $$\gamma$$ can be tuned to either suppress ($$\gamma>1$$) or enhance ($$\gamma<1$$) the contribution of low centrality nodes in determining the centrality of the layers in which they are active. For $$s=-1$$ however, layers have larger influence if they include fewer highly influential nodes. In other words this algorithm awards elite layers. In this case the parameter $$\gamma$$ can be tuned to either enhance ($$\gamma>1$$) or suppress ($$\gamma<1$$) the contribution of low centrality nodes in determining the centrality of the layers in which they are active. For $$a=0$$ and $$\gamma=0$$ the influences of the layers are all equal and the MultiRank reduces to the Pagerank of the aggregated network. For all the other possible values of the parameters the MultiRank is not reducible to the PageRank of the aggregated network. We implemented the MultiRank algorithm in a MATLAB code that is publicly available [47]. The code has a computational complexity $${\mathcal O}(N^2M)$$ as tested on an ensemble of multiplex networks formed by uncorrelated Poisson layers. 2.3 Possible variations of the MultiRank algorithm The MultiRank algorithm, is fully defined by the Eqs. (12) and (13). However this algorithm can be modified by changing the first equation (i.e. Eq. (12)) determining the centrality of the nodes. In fact instead of adopting a PageRank algorithm for the centrality of the nodes, it is possible to consider either the eigenvector centrality or the Katz centrality [2]. In the first case, Eq. (12) could be substituted by \begin{eqnarray} \lambda_1X_i&=&\sum_{j=1}^N G_{ji}X_j,\nonumber \\ \sum_{i=1}^NX_i&=&1, \end{eqnarray} (14) where $$\lambda_1$$ is the largest eigenvalue of $${\bf G}$$. In the second case, Eq. (12) could be substituted by \begin{align} X_i=&\hat{\alpha} \sum_{j=1}^N {G_{ji}}X_j+ v_i, \end{align} (15) where $$\hat{\alpha}>0$$ is suitably chosen to ensure the convergence of the algorithm and $$v_i$$ is defined in Eq. (5). 3. Application to real data sets In this section, we will consider the application of the MultiRank algorithm defined in Section 2.2 to three main examples of multiplex data sets: the European Air Transportation Multiplex Network [31], the Pierre Auger Multiplex Collaboration Network [34] and the FAO Multiplex Trade Network [30]. All data sets are publicly available at the website [48]. Table 1 summarizes the main properties of the three data sets and the number of iterations necessary for convergence of the MultiRank algorithm. TABLE 1 Main properties of the three analysed data sets: the European Air Transportation Multiplex Network (EA) [31], the Pierre Auger Multiplex Collaboration Network (PA) [34] and the FAO Multiplex Trade Network (FAO) [30]. The quantities specified are $$N$$ the total number of nodes, $$M$$ the total number of layers and $$I$$ the number of iterations necessary for a convergence of the MultiRank algorithm within a tolerance of $$10^{-4}$$ on the square error of the vector of node centralities $$X_i$$. Since for every value of the parameters the number of iterations $$I$$ can vary we have indicated the range of variability of $$I$$ for each one of the three data sets. Additionally, we have indicated if the network is directed and/or weighted Data set N M I Directed Weighted EA 450 37 20–60 No No PA 514 16 42–44 No No FAO 214 364 16–18 Yes Yes Data set N M I Directed Weighted EA 450 37 20–60 No No PA 514 16 42–44 No No FAO 214 364 16–18 Yes Yes TABLE 1 Main properties of the three analysed data sets: the European Air Transportation Multiplex Network (EA) [31], the Pierre Auger Multiplex Collaboration Network (PA) [34] and the FAO Multiplex Trade Network (FAO) [30]. The quantities specified are $$N$$ the total number of nodes, $$M$$ the total number of layers and $$I$$ the number of iterations necessary for a convergence of the MultiRank algorithm within a tolerance of $$10^{-4}$$ on the square error of the vector of node centralities $$X_i$$. Since for every value of the parameters the number of iterations $$I$$ can vary we have indicated the range of variability of $$I$$ for each one of the three data sets. Additionally, we have indicated if the network is directed and/or weighted Data set N M I Directed Weighted EA 450 37 20–60 No No PA 514 16 42–44 No No FAO 214 364 16–18 Yes Yes Data set N M I Directed Weighted EA 450 37 20–60 No No PA 514 16 42–44 No No FAO 214 364 16–18 Yes Yes 3.1 European air transportation multiplex network The European Air Transportation Multiplex Network [31] is a multiplex network data set comprising $$M=37$$ layers (European airline companies) and $$N=450$$ nodes (European airports). Each layer is unweighted and undirected and is formed by the flight connections operated by a given European airline company. The data set is widely used in the context of multiplex network research as one of the most interesting, clean, well defined and freely available data sets where to test new measures and algorithms for the extraction of relevant information. Here we discuss the application of the MultiRank algorithm to this data set. Therefore our goal will be to rank simultaneously European airports and airline companies. In Figs 2 and 3, we show a map of Europe where different airports are assigned the MultiRank centrality for different values of the parameters $$s, a$$ and $$\gamma$$. The distribution of the node centralities tends to become significantly more skewed for larger values of $$\gamma$$. In order to capture this phenomena in Fig. 4, we report the distribution $$P(X)$$ of the airports MultiRank centralities $$X$$ for the same parameters used in Figs 2 and 3. Fig. 2. View largeDownload slide The maps representing the centrality $$X_i$$ of the European airports in the European Air Transportation Multiplex Network according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$; $$a=1$$ and $$\gamma=1,3$$. In the case $$s=1, a=1,$$ we observe that the centrality of the nodes are more heterogeneously distributed for $$\gamma=3$$ than for $$\gamma=1$$ with few airports acquiring a centrality much higher than the others. Fig. 2. View largeDownload slide The maps representing the centrality $$X_i$$ of the European airports in the European Air Transportation Multiplex Network according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$; $$a=1$$ and $$\gamma=1,3$$. In the case $$s=1, a=1,$$ we observe that the centrality of the nodes are more heterogeneously distributed for $$\gamma=3$$ than for $$\gamma=1$$ with few airports acquiring a centrality much higher than the others. Fig. 3. View largeDownload slide The maps representing the centrality $$X_i$$ of the European airports in the European Air Transportation Multiplex Network according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$; $$a=0$$ and $$\gamma=1,3$$. As for $$s=1$$ and $$a=1$$ (maps reported in Fig. 2), we observe that for $$\gamma=3$$ the centralities of the nodes are more heterogeneously distributed than for $$\gamma=1$$. Fig. 3. View largeDownload slide The maps representing the centrality $$X_i$$ of the European airports in the European Air Transportation Multiplex Network according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$; $$a=0$$ and $$\gamma=1,3$$. As for $$s=1$$ and $$a=1$$ (maps reported in Fig. 2), we observe that for $$\gamma=3$$ the centralities of the nodes are more heterogeneously distributed than for $$\gamma=1$$. Fig. 4. View largeDownload slide Distribution $$P(X)$$ of the MultiRank centralities $$X$$ of the airports in the European Air Transportation Multiplex Network for the same parameters $$s=1,-1$$, $$a=0,1$$ and $$\gamma=1,3$$ considered in Figs 2 and 3. Fig. 4. View largeDownload slide Distribution $$P(X)$$ of the MultiRank centralities $$X$$ of the airports in the European Air Transportation Multiplex Network for the same parameters $$s=1,-1$$, $$a=0,1$$ and $$\gamma=1,3$$ considered in Figs 2 and 3. In Figs 5 and 6, we plot respectively the rank of the top 15 European airline companies and of the rank of the top 15 European airports according to the MultiRank evaluated for $$s=1,-1$$ and $$a=0,1$$. For the relevant case $$s=-1,a=1$$ we observe a stable ranking of the top 4 ranked airline companies as a function of $$\gamma$$ given by Ryanair, Easyjet, Lufthansa, Air Berlin and rather stable ranking of airports with Stansted airport (STN) remaining the rank one airport for all the considered range of values of $$\gamma$$. When we calculate the MultiRank algorithm for different values of the parameter $$s$$ and $$a$$, with $$(s,a)\neq (-1,1)$$ we note that the algorithm experience an instability for $$\gamma=\gamma^{\star}>1$$. Fig. 5. View largeDownload slide Ranking of the top 15 European airline companies in the European Air Transportation Multiplex Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1 $$ and $$a=0,1$$ as a function of $$\gamma\in (0,3)$$. Fig. 5. View largeDownload slide Ranking of the top 15 European airline companies in the European Air Transportation Multiplex Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1 $$ and $$a=0,1$$ as a function of $$\gamma\in (0,3)$$. Fig. 6. View largeDownload slide Ranking of the top 15 European airports in the European Air Transportation Multiplex Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=0,1$$ as a function of $$\gamma\in (0,3)$$. Fig. 6. View largeDownload slide Ranking of the top 15 European airports in the European Air Transportation Multiplex Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=0,1$$ as a function of $$\gamma\in (0,3)$$. The instability observed for $$s=1$$ at large values of $$\gamma$$ can be explained as the result of the suppressed role that low-rank airports have in determining the centrality of the layers in which they are active. For instance, this observation well describes the fact that for $$s=1, a=1$$ a low cost airline (such as Ryanair) that typically serve either its own hub airports or much smaller airports, decrease its rank for larger value of $$\gamma$$ to the advantage of a major carrier such as Lufthansa. On the contrary, for $$s=-1$$ at large value of $$\gamma$$ low rank airports acquire a major role in determining the centrality of the layers in which they are active. This effect is even more pronounced for $$a=0$$ when the influence of the airline companies (layers) is not proportional to the number of flight connections. This explains for instance that for $$s=-1,a=0$$ Lufthansa decreases its rank as $$\gamma$$ increases. The maps reported in Fig. 3 showing the MultiRank centrality of the airports for different MultiRank parameters describe visually the effect of the instability observed for $$(s,a)\neq (-1,1)$$. For $$\gamma=3$$, after these instabilities have taken place, we observe that few airports acquire a centrality much higher than all the other airports. For instance in the case $$s=1, a=1$$ well before the instability, for $$\gamma=1$$, the airline Ryanair and its major airport STN are on the top of the rank, for $$\gamma=3$$, above the instability, we observe the two major Lufthansa airports, that is Munich Airport (MUC) and Frankfurt Airport (FRA), at the top of the airport rankings acquiring a centrality much higher than all the other European airports. 3.2 Pierre Auger Multiplex Collaboration Network The Pierre Auger Multiplex Collaboration Network [34] describes the scientific collaboration networks established in the framework of the Pierre Auger experiment. The Pierre Auger experiment [49] aims at constructing a large array of detectors for studying cosmic rays. The Pierre Auger Multiplex Collaboration Network is formed by $$M=16$$ layers (scientific topics) and $$N=514$$ nodes (scientists). Each layer is a network characterizing the collaborations in a given scientific topic. Each layer of this multiplex network is undirected and unweighted. The names of the nodes (scientists) have been anonymized while the names of the layers are available. These layers are: Neutrinos, Detector, Enhancements, Anisotropy, Point-source, Mass-composition, Horizontal, Hybrid-reconstruction, Spectrum, Photons, Atmospheric, SD-reconstruction, Hadronic-interactions, Exotics, Magnetic and Astrophysical-scenarios. We have run the MultiRank algorithm on this data set ranking simultaneously nodes and layers. However in Fig. 7, we display only the results relative to the layers due to the fact that the node labels and identities are not disclosed. Fig. 7. View largeDownload slide Ranking of the top 15 topics in the Pierre Auger Multiplex Collaboration Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=0,1$$ as a function of $$\gamma\in (0,3)$$. Fig. 7. View largeDownload slide Ranking of the top 15 topics in the Pierre Auger Multiplex Collaboration Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=0,1$$ as a function of $$\gamma\in (0,3)$$. The MultiRank results clearly establish the importance of experimental advances for the Pierre Auger Multiplex Collaboration Network. We observe that for each of the cases $$s=1,-1$$ and $$a=1,0$$ the topic Enhancements is at the top of the ranking. Its position is actually stable in every case with the exception of the case $$s=1, a=0$$. For cases $$a=1$$ we observe in second position (rank 2) the topic Detector both in case $$s=1$$ and in case $$s=-1$$. Overall, the ranking of the layers seems to be more stable for the cases $$a=1$$ than for the cases $$a=0$$. Notably, while running the MultiRank on this data set we have not detected any instability in this case. 3.3 FAO Multiplex Trade Network As a third example of a multiplex network, we have considered here the FAO Multiplex Trade Network [30] collecting data coming from the FAO of the United Nations. This is a major example of a trade network between countries. Other similar multiplex trade networks including also other types of products have been extensively studied in the literature [27–29, 50]. This data set includes $$N=214$$ nodes (countries) and $$M=364$$ layers (food and agriculture products). Each layer is weighted and directed characterizing which country is importing from which country a given product, and indicating the amount of the imported values in US dollars. Therefore the network is in this case both directed and weighted. We characterized the MultiRank of this data set by ranking simultaneously countries and products. Note that given the definition of MultiRank we are taking into account not only the bipartite nature of the graph determining which country exports which product as in Refs [7–9] but instead we consider the overall multiplex network structure of the FAO Multiplex Trade Network. In fact, we process the information of which country is trading a given product with which other country. In this case, it is natural to focus our attention on the MultiRank algorithm with $$a=1$$, since the importance of a product should be clearly related to the total amount in US dollars of its international trade. In Figs 8 and 9, we report respectively the rank of the top 15 countries and 15 products as a function of $$\gamma\in (0,3)$$ for different values of the parameter $$s$$, that is for $$s=1$$ and $$s=-1$$, and for $$a=1$$. Fig. 8. View largeDownload slide Ranking of the top 15 countries in the FAO Multiplex Trade Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=1$$ as a function of $$\gamma\in (0,3)$$. Fig. 8. View largeDownload slide Ranking of the top 15 countries in the FAO Multiplex Trade Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=1$$ as a function of $$\gamma\in (0,3)$$. Fig. 9. View largeDownload slide Ranking of the top 15 products in the FAO Multiplex Trade Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=1$$ as a function of $$\gamma\in (0,3)$$. Fig. 9. View largeDownload slide Ranking of the top 15 products in the FAO Multiplex Trade Network (listed from top to bottom in order of decreasing centrality) according to the MultiRank algorithm are here shown for different values of the parameters $$s=-1,1$$ and $$a=1$$ as a function of $$\gamma\in (0,3)$$. We observe a very stable list of top-ranked countries: USA, China, Germany, Japan, China Mainland and France both for $$s=1$$ and for $$s=-1$$. For the products, we observe in top rank positions a composition of prime necessity products such as Soybeans and Crude-materials and of more expensive products which are in high demand such as Wine and Cocoa-beans. 4. Comparison with other rankings The MultiRank algorithm is a novel centrality measure that unlike previously existing algorithms does not only rank nodes but also ranks the layers of a multiplex network. Also in the case in which we just focus on the ranking of the nodes, as long as $$\gamma>0$$ the MultiRank is not reducible to previously considered rankings. This is a desirable property because it implies that the MultiRank is an independent centrality measure. However, it is desirable to test whether the MultiRank it a meaningful centrality measure and whether it is in line with other observations and indicators of the centrality of the nodes. A way to check the consistency of the results obtained with the MultiRank algorithm is to measure the correlations between the MultiRank of the nodes and the ranking of the nodes provided by some other independent measure. For the European Air Transportation Multiplex Network, we have considered the correlations between the MultiRank of the airports and their ranking according to their total number of passengers. The data on the total number of passengers for each European airport has been taken from the Eurostat webpage [51] for the same year (2011) in which the European Air Multiplex Transportation Network has been collected. The ranking of the top 100 airports with largest number of passengers is compared with their MultiRank. The Spearman rank correlations between the rankings is shown in Fig. 10A and is very significant as the $$P$$-value ranges between $$1\times 10^{-17}$$ and $$5\times 10^{-17}$$. Fig. 10. View largeDownload slide In panel A, the Spearman rank correlation $$S$$ between the ranking of the top 100 airports with largest number of passengers and their MultiRank is shown as a function of $$\gamma$$ for the parameter values $$s=1,-1$$ and $$a=0,1$$. The solid dashed line in panel A indicates the Spearman rank correlation between the ranking of the airports determined by the number of passengers and the ranking provided by the PageRank of the aggregated network. In panel B, the Spearman rank correlation $$S$$ between the ranking of the top 100 countries with largest GDP and their MultiRank is shown as a function of $$\gamma$$ for the parameter values $$s=1,-1$$ and $$a=0,1$$. The solid dashed line in panel B indicates the Spearman rank correlation between the ranking determined by the GDP and the ranking provided by the PageRank of the aggregated network. Fig. 10. View largeDownload slide In panel A, the Spearman rank correlation $$S$$ between the ranking of the top 100 airports with largest number of passengers and their MultiRank is shown as a function of $$\gamma$$ for the parameter values $$s=1,-1$$ and $$a=0,1$$. The solid dashed line in panel A indicates the Spearman rank correlation between the ranking of the airports determined by the number of passengers and the ranking provided by the PageRank of the aggregated network. In panel B, the Spearman rank correlation $$S$$ between the ranking of the top 100 countries with largest GDP and their MultiRank is shown as a function of $$\gamma$$ for the parameter values $$s=1,-1$$ and $$a=0,1$$. The solid dashed line in panel B indicates the Spearman rank correlation between the ranking determined by the GDP and the ranking provided by the PageRank of the aggregated network. For the FAO Multiplex Trade Network, it is possible to compare the MultiRank of the countries with their ranking according to the GDP. The data of the GDP of the countries has been collected from the World Bank database [52] for the same year (2010) in which the FAO Multiplex Trade Network has been collected. The ranking of the top 100 countries with largest GDP is compared with their MultiRank. The Spearman rank correlations between the rankings shown in Fig. 10B are very significant and correspond to a $$P$$-value ranging between $$2\times 10^{-27}$$ and $$2\times 10^{-14}$$ depending on the MultiRank parameters. Therefore in both cases we observe that the ranking provided by the MultiRank is in line with other independent observations. Additionally, we observe that the Spearman rank correlations are in some cases higher than the Spearman correlations calculated using instead of the MultiRank the PageRank of the aggregated network. In both cases, we observe that the MultiRank with $$s=-1$$ and $$a=0$$ achieves higher Spearman correlations. However, in none of the considered parameter values the ranking provided by MultiRank is reducible to the rank obtained by the passenger number or the GDP showing that the MultiRank is an independent measure of centrality. 5. Conclusions In this article, we have presented the MultiRank algorithm for ranking nodes and layers in large multiplex networks. The ranking of nodes and layers is determined by a coupled set of equations. The centrality of the nodes is evaluated according to a PageRank algorithm based on a weighted random walk where the probability to hop to a neighbour node of a given layer $$\alpha$$ depends on the influence (centrality) of the layer. Vice-versa, the influences of the layers are determined by the centrality of the nodes that are active on it. The equations determining the influence of the layers exploit the bipartite structure of the network between nodes and layers that can be constructed starting from the complete information about the multiplex structure. The MultiRank depends on two main parameters $$s$$ and $$a$$. The parameter $$a$$ allows to attribute to layers an influence proportional to their total weight or alternatively to attribute an influence that does not take into account the total weight of the layers. The parameter $$s$$ instead, allows us to attribute more influence to more popular layers or to elite layers. The proposed algorithm is very flexible and is able to efficiently rank multiplex networks with many layers. It can be applied to undirected and directed multiplex networks with weighted or unweighted links. We have shown that the MultiRank is able to extract relevant information from the data sets of three major examples of multiplex networks (including the European Air Transportation Multiplex Network, the Pierre Auger Multiplex Collaboration Network and the weighted and directed FAO Trade Multiplex Network). In conclusion, this algorithm provides a framework to evaluate the centrality of nodes and layers in multiplex networks. 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This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
Journal of Complex Networks – Oxford University Press
Published: Oct 1, 2018
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