# Corner Centrality of Nodes in Multilayer Networks: A Case Study in the Network Analysis of Keywords
## 1 Introduction  
Which countries are most relevant in the world for being the largest producers of raw materials (food, minerals, etc.) for the rest of the world? What are the most effective drugs for a set of diseases? Which diplomats are the most relevant for carrying out deals between countries in conflict? Finally, given a set of scientific articles, which keywords are the most relevant? These questions are examples of where we need to discover the most relevant agents (i.e., countries, drugs, diplomatic persons, or keywords) in a complex system composed of agents and the relations between them. These complex systems often use networks to represent these relations. To this aim, network theory is an important area that offers solid tools for describing the complex system in different environments such as biology, social networks, information technology, and engineering [1]. Most of these complex systems use graphs to represent these relations and the characteristics between the entities representing the system. Many problems have historically been modeled with simple graphs, e.g., the traveling salesman problem [2], minimum spanning tree [3], etc. Therefore, these simple situations need only be represented with a single graph. However, complex systems need more than one graph to be properly represented, and in many cases, a multilayer network is used for these representations. A multilayer network is made up of a set of layers, each one represented by a graph [4; 5]. The nodes in a multilayer network can have different states. For example, a person can be analyzed by her/his friendships, jobs, or relationships in different social networks. Thus, a layer can be described as a set of state nodes and the edges between these state nodes [6]. An initial idea that could be used to represent these complex systems would be to break the system down into independent graphs and analyze each graph. However, not taking the potential dependency into consideration for these graphs can lead to a cascade of failures and a misinterpretation of the system's reality [7]. If the system is represented by a multilayer network composed of a set of layers (with each layer representing a portion of the system's information using a graph) then that allows us to describe intrac relationships between members of the same layer as inter-relationships between members of different layers. Thus, the multilayer network may have intralinks and interlinks between network nodes. For example, in Figure 1, Multilayer Network 2 (MLN2) contains only intralinks, while MLN3 has intralinks and interlinks connecting nodes between the layers L1 and L2. Figure 1: Examples of different multilayer networks. MLN1 is a multilayer network with two layers that have the same nodes and links. However, the nodes in each layer have different levels of importance. MLN2 has two layers with the same nodes but different links, and the nodes have different levels of importance. MLN3 has two layers with different nodes and links and the two layers are connected by links. MLN4 has three layers. Multilayer networks are used to represent complex systems, such as in traffic control, social networks, or biological systems [7; 8]. Traffic control systems can describe traffic dynamics [9], for example, when passengers are transported by public transport. Every layer can represent the ways in which citizens travel, such as by bus, subway, or tram, in a city. Additionally, in air traffic, the layers correspond to flight routes operated by different airline companies. Recent works, for example, use complex network to analyze the role of conformist and profiteeer players in evolutionary games [10], and to simulate how asymptomatic people can spread COVID-19 with a high level of accuracy [11]. For urban networks, in [12], the PageRank algorithm (APA) was adapted, providing a model with which to establish a ranking of nodes in spatial networks according to their importance within them.