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The current flow betweenness centrality is a useful tool to estimate traffic status in spatial networks and, in general, to measure the intermediation of nodes in networks where the transition between them takes place in a random way. The main drawback of this centrality is its high computational cost, especially for very large networks, as it is the case of urban networks. In this paper, a new approach to the current flow betweenness centrality for its practical application in urban networks with data is presented and discussed. The new centrality measure allows the estimation of pedestrian flow developed in urban networks, taking into account both the network topology and its associated data. In addition, its computational cost makes it suitable for application in networks with a large number of nodes. Some examples are studied in order to better understand the characteristics and behaviour of the proposed centrality in the context of the city. ; Partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P .

Networks are useful to describe the structure of many complex systems. Often, understanding these systems implies the analysis of multiple interconnected networks simultaneously, since the system may be modelled by more than one type of interaction. Multiplex networks are structures capable of describing networks in which the same nodes have different links. Characterizing the centrality of nodes in multiplex networks is a fundamental task in network theory. In this paper, we design and discuss a centrality measure for multiplex networks with data, extending the concept of eigenvector centrality. The essential feature that distinguishes this measure is that it calculates the centrality in multiplex networks where the layers show different relationships between nodes and where each layer has a dataset associated with the nodes. The proposed model is based on an eigenvector centrality for networks with data, which is adapted according to the idea behind the two-layer approach PageRank. The core of the centrality proposed is the construction of an irreducible, non-negative and primitive matrix, whose dominant eigenpair provides a node classification. Several examples show the characteristics and possibilities of the new centrality illustrating some applications. ; This research is partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P.

Diversity is an important measure that according to the context, can describe different concepts of general interest: competition, evolutionary process, immigration, emigration and production among others. It has been extensively studied in different areas, as ecology, political science, economy, sociology and others. The quality of spatial context of the city can be gauged through this measure. The spatial context with its corresponding dataset can be modelled using spatial networks. Consequently, this allows us to study the diversity of data present in this specific type of networks. In this paper we propose an algorithm to measure diversity in spatial networks based on the topology and the data associated to the network. In the experiments developed with networks of different sizes, it is observed that the proposed index is independent of the size of the network, but depends on its topology. ; Partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P.

The Adapted PageRank Algorithm (APA) proposed by Agryzkov et al. provides us a method to establish a ranking of nodes in an urban network. We can say that it constitutes a centrality measure in urban networks, with the main characteristic that is able to consider the importance of data obtained from the urban networks in the process of computing the centrality of every node. Starting from the basic idea of this model, we modify the construction of the matrix used for the classification of the nodes in order of importance. In the APA model, the data matrix is constructed from the original idea of PageRank vector, given an equal chance to jump from one node to another, regardless of the topological distance between nodes. In the new model this idea is questioned. A new matrix with the data network is constructed so that now the data from neighboring nodes are considered more likely than data from the nodes that are farther away. In addition, this new algorithm has the characteristic that depends on a parameter α, which allows us to decide the importance attached, in the computation of the centrality, to the topology of the network and the amount of data associated with the node. Various numerical experiments with a network of very small size are performed to test the influence of the data associated with the nodes, depending always on the choice of the parameter α. Also we check the differences between the values produced by the original APA model and the new one. Finally, these measures are applied to a real urban network, in which we perform a visual comparison of the results produced by the various measures calculated from the algorithms studied. ; Partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2014-53855-P.

Identifying the influential nodes in complex networks is a fundamental and practical topic at the moment. In this paper, a new centrality measure for complex networks is proposed based on two contrasting models that have their common origin in the well-known PageRank centrality. On the one hand, the essence of the model proposed is taken from the Adapted PageRank Algorithm (APA) centrality, whose main characteristic is that constitutes a measure to establish a ranking of nodes considering the importance of some dataset associated to the network. On the other hand, a technique known as two-layers PageRank approach is applied to this model. This technique focuses on the idea that the PageRank centrality can be understood as a two-layer network, the topological and teleportation layers, respectively. The main point of the proposed centrality is that it combines the APA centrality with the idea of two-layers; however, the difference now is that the teleportation layer is replaced by a layer that collects the data present in the network. This combination gives rise to a new algorithm for ranking the nodes according to their importance. Subsequently, the coherence of the new measure is demonstrated by calculating the correlation and the quantitative differences of both centralities (APA and the new centrality). A detailed study of the differences of both centralities, taking different types of networks, is performed. A real urban network with data randomly generated is evaluated as well as the well-known Zachary's karate club network. Some numerical results are carried out by varying the values of the α parameter—known as dumping factor in PageRank model—that varies the importance given to the two layers (topology and data) within the computation of the new centrality. The proposed algorithm takes the best characteristics of the models on which it is based: on the one hand, it is a measure of centrality, in complex networks with data, whose calculation is stable numerically and, on the other hand, it is able to separate the topological properties of the network and the influence of the data. ; Partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P.

Cities can be represented by spatial networks, and the mathematical structure that defines a spatial network is a graph. Taking into account this premise, this paper is focused on analysing information on an urban scale by combining a new ray-casting visibility index with a data set of the urban street network. The visibility index provides information about the most visible buildings or areas. We relate this index with other data extracted from the city, with the aim of generating and analysing information about urban elements. To corroborate this idea, real data are analysed. The dataset is related to the heritage conservation of the buildings of the Villaflora suburb, located in the city of Quito (Ecuador). This information is processed, together with the visibility index, with the aim of determining the conservation degree of the urban areas most visually exposed to pedestrians or visitors. The combination of both values—heritage conservation and visibility index—is carried out by means of two new indices, IP and IN , which are defined using two-variable exponential functions. ; This research is partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P.

In about 1940, Quito's urban planning department contemplated the creation of a new suburb called Villaflora following the garden city model: homes in connection with nature but also near services. In Villaflora we do not find monumental elements that characterize patrimonial architecture; the value of Villaflora's patrimony is in its urban model characterized by some architectonic elements. However, Villaflora is valuable because it is the result of a unique urban model. Over the years, the suburb has suffered profound degradation from the point of view of its patrimonial conservation. Hence, we propose an urban intervention in the suburb that contemplates the restoration of some important elements in the urban layout, without altering the commercial structure of the same. To accomplish this task we perform a study of the heritage conservation of each of the buildings of the suburb, as well as a study of the commercial activity that is developed in the suburb in order to determine those areas with the highest commercial activity and as a consequence, a greater presence of people in the streets and public spaces. ; Partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2014-53855-P.

A city may be represented by spatial networks. The mathematical structure that defines a spatial network is a graph, which is constructed from a set of nodes and a set of edges. According to the meaning we assign to nodes and edges, different types of graphs describing urban networks may be obtained, each of them with their special characteristics. In this paper, we study five different types of graphs associated to the spatial network of the urban layout. They are briefly described and compared pointing out their main features. A case study is performed using the urban network of Alicante, Spain. By means of the study of the distribution of degrees of each one of the graphs obtained experimentally certain conclusions are obtained about the morphology of the urban network itself. ; This work was partially supported by the Spanish Government, Ministerio de Economía y Competividad, which reference number is TIN2014-53855-P.

Usually, the nodes' interactions in many complex networks need a more accurate mapping than simple links. For instance, in social networks, it may be possible to consider different relationships between people. This implies the use of different layers where the nodes are preserved and the relationships are diverse, that is, multiplex networks or biplex networks, for two layers. One major issue in complex networks is the centrality, which aims to classify the most relevant elements in a given system. One of these classic measures of centrality is based on the PageRank classification vector used initially in the Google search engine to order web pages. The PageRank model may be understood as a two-layer network where one layer represents the topology of the network and the other layer is related to teleportation between the nodes. This approach may be extended to define a centrality index for multiplex networks based on the PageRank vector concept. On the other hand, the adapted PageRank algorithm (APA) centrality constitutes a model to obtain the importance of the nodes in a spatial network with the presence of data (both real and virtual). Following the idea of the two-layer approach for PageRank centrality, we can consider the APA centrality under the perspective of a two-layer network where, on the one hand, we keep maintaining the layer of the topological connections of the nodes and, on the other hand, we consider a data layer associated with the network. Following a similar reasoning, we are able to extend the APA model to spatial networks with different layers. The aim of this paper is to propose a centrality measure for biplex networks that extends the adapted PageRank algorithm centrality for spatial networks with data to the PageRank two-layer approach. Finally, we show an example where the ability to analyze data referring to a group of people from different aspects and using different sets of independent data are revealed. ; This research is partially supported by the Spanish Government, Ministerio de Economía y Competividad, grant number TIN2017-84821-P.

[EN] Usually, the nodes' interactions in many complex networks need a more accurate mapping than simple links. For instance, in social networks, it may be possible to consider different relationships between people. This implies the use of different layers where the nodes are preserved and the relationships are diverse, that is, multiplex networks or biplex networks, for two layers. One major issue in complex networks is the centrality, which aims to classify the most relevant elements in a given system. One of these classic measures of centrality is based on the PageRank classification vector used initially in the Google search engine to order web pages. The PageRank model may be understood as a two-layer network where one layer represents the topology of the network and the other layer is related to teleportation between the nodes. This approach may be extended to define a centrality index for multiplex networks based on the PageRank vector concept. On the other hand, the adapted PageRank algorithm (APA) centrality constitutes a model to obtain the importance of the nodes in a spatial network with the presence of data (both real and virtual). Following the idea of the two-layer approach for PageRank centrality, we can consider the APA centrality under the perspective of a two-layer network where, on the one hand, we keep maintaining the layer of the topological connections of the nodes and, on the other hand, we consider a data layer associated with the network. Following a similar reasoning, we are able to extend the APA model to spatial networks with different layers. The aim of this paper is to propose a centrality measure for biplex networks that extends the adapted PageRank algorithm centrality for spatial networks with data to the PageRank two-layer approach. Finally, we show an example where the ability to analyze data referring to a group of people from different aspects and using different sets of independent data are revealed. ; This research is partially supported by the Spanish Government, ...