**3. Results**

In this section, an example of a biplex network is analyzed in detail in order to highlight the possibilities offered by disposing of a centrality measure that establishes a classification of the nodes in order of importance.

For this purpose, let us consider a graph of 20 nodes, where each node represents a physical person; specifically, a player from a football team. Around this group the example is developed.

With these 20 nodes, let us proceed to construct a network with two layers that relate the nodes in a different way, taking two datasets (one for each of the layers). The calculation of the APABI centrality on this biplex network will allow us to determine the importance of each member of the team and obtain a classification of them in order of importance.

First, a layer *l*1 is constructed with the 20 nodes where the relationships between the members of the team are analyzed from the point of view of social or virtual relationships. Thus, an undirected graph is constructed with an adjacency matrix *A*1 in which two nodes are joined by an edge if they are related or linked through a social network. The graph of social relations between the nodes is shown in Figure 2. The data that are considered in this layer associated with each node are related to the number of messages that each person receives from their teammates in a period of time.

**Figure 2.** Graphs of the first layer (**left**) and the second one (**right**).

Secondly, a second layer *l*2 is constituted from the same 20 nodes of layer *l*1, but analyzing in *l*2 how the players relate to each other within the game, that is, if they associate with each other in the game. Thus, two players who combine with each other or pass the ball with some assiduity during a match are connected by an edge. From these links, it is possible to build a new adjacency matrix associated with this layer, which we denote by *A*2. In most team sports with a ball, each player occupies a specific position in the field of play, covering a certain area. Players (nodes) that occupy closer positions associate or relate more easily with each other than with those who are further away. For example, a defender is associated more with a midfielder than with a forward. The graph of game relations between the nodes is shown in Figure 2.

Both the links and the data associated to each of the 20 nodes of the graph are summarized in Table 1. There, the second column specifies the social links of every node while in the four column the game links between them are detailed. So, for example, the table shows that node 1 (player labeled as 1) has social interactions with the nodes {2, 5, 7, 9, 16, 17, 19, 20} while presents strong interactions within the game with the nodes {2, 4, 5, 6, 9, 12, 13, 14, 18, <sup>19</sup>}.


**Table 1.** Data associated to the biplex network constructed from the team.

Datasets about the number of messages received during one day through social networks and the number of games that have played along the season are detailed in columns three and five, respectively. So, node 1 has received 15 messages in a day and has played 33 games in the season.

In this example, one of the advantages of working with biplex networks becomes manifest, such as the possibility of studying different relationships between the same set of nodes, analyzing the correlation between them. This example shows the advantage offered by adding what we can call a data layer in each of the multilayers of the network. We can assign the data that we consider appropriate to the specific relationships that we are representing by means of the corresponding graph. Thus, as can be seen in this example, in the layer where the social relations between the nodes are considered, we introduce the data corresponding to the number of received messages. However, in the second layer where game relations are represented, data are completely different, since now the number of games played are considered. Thus, each layer allows us to introduce one or more data sets related to the relationships of the nodes. This leads us to affirm that the inclusion of data in each of the layers enriches the nature of the problems that can be analyzed.

The objective in this example is to determine the most important or influential players within the team. For this task, two different aspects may be evaluated; on the one hand, the importance of the nodes from the point of view of the social relations that are established between them through messages, social networks, or any other virtual means. The nodes that have an intense social activity in the group create a very important union within the group, being very influential for other nodes. On the other hand, the importance of the nodes from the point of view of the game may be also evaluated, that is, which players are more important in the game, for their participation or quality. In other words, it is decisive to look for the leaders of the group, analyzing their importance from the social and technical point of view.

In order to determine the importance of the nodes of the biplex network of this example, the APABI centrality described in Section 2.3 has been calculated. Algorithm 2 has been executed using the information shown in Table 1. The numerical results shown in Table 2 are graphically displayed in Figure 3. This calculation gives us the importance of the nodes relating both layers. Figure 4 shows the final graph considering the information of two layers and the final result of the APABI centrality for each node, representing the size of each node according to its importance.


**Table 2.** Adapted PageBreak algorithm (APA) centrality for layers *l*1 and *l*2 and APA biplex (APABI) centrality for biplex network.

**Figure 3.** Centralities shown in Table 2.

**Figure 4.** Adapted PageRank algorithm biplex (APABI) centrality for the example object of this study.

The APABI centrality shows that the nodes that can be classified as the most important, the leaders within the group, are nodes 20 and 18, respectively. Note that node 20, which is the most important, is not the node that receives the most messages from its colleagues, being the node 18 is the one that receives the most messages from his teammates, but is second in the ranking.

Finally, it should be noted that, as discussed in Section 2.4, the use of the power method to calculate the stationary vector of the Markov chain that forms the stochastic matrix *MBI* provides the numerical stability needed in the implemented algorithm.

We have performed several tests with randomly generated adjacency matrices of different sizes in a range from 10 to 10,000 and we have obtained stable results. Matrices of dimension exceeding 10<sup>5</sup> cannot be stored in the central memory of most computers, except for sparse matrices. Consequently, the only matrix-arithmetic operation that is easily performed is a matrix-vector product. This makes possible to use this centrality algorithm for large matrices. In the scope of our research with urban network matrices, the sizes of the case studies are relatively large, with 2000–5000 nodes, approximately.
