José-Francisco Vicent

An algorithm for ranking the nodes of an urban network based on the concept of PageRank vector

Abstract This paper presents a new method to establish a ranking of nodes in an urban network. In the original PageRank algorithm, a single PageRank vector is computed to determine the relative importance of Web pages, independent of any particular search query. We follow a similar reasoning, adapting the concept of PageRank vector to urban networks. We translate an urban network to graph theory language, where the nodes represent crossings or squares and the edges represent the connections between nodes. In this scenario, our main goal is to establish a ranking of importance of the different nodes in the graph. Unlike the PageRank model which only takes into account the connections between the Web pages, in our method we must consider other external factors to carry out the classification. These external factors may have some characteristics associated with the nodes and they are different according to the problem we are working. These characteristics are usually related to the presence of some facilities or endowments in the nodes of the network, like for example the presence of restaurants, bars, shopping centers, stores, and so on. A data matrix will collect the quantification of each of the characteristics studied for each node and will play an important role in the process of classification. Considering the influence of all these characteristics, we construct a matrix with some interesting algebraic features which allow us to compute an eigenvector associated to the dominant eigenvalue λ = 1 . This vector constitutes the solution to our problem of ranking the nodes of the network. The model is applied to a real example, in which we consider a part (the old town) of the city of Murcia (Spain). For this example, we apply the PageRank model, as well as the proposed model in this paper, in order to perform a comparative study of both models. This example clearly shows the importance of considering external factors to quantify the urban network nodes.

Analysis and design of a secure key exchange scheme

We propose a new key exchange scheme where the secret key is obtained by multiplying the powers of block upper triangular matrices. After studying the cryptographic properties of these block matrices, the theoretical aspects of this scheme are analyzed, concluding that common ciphertext attacks are not applicable to this cryptosystem. Moreover, our proposal is compared with the Diffie-Hellman scheme achieving satisfactory results.

A nonlinear elliptic curve cryptosystem based on matrices

We propose a new mathematical problem that is applicable to public key cryptography. Based on the Discrete Logarithm Problem (DLP), it uses certain elements formed by two matrices with elements in a finite field and a matrix whose elements are points of an elliptic curve. With this system, we get a larger key space without increasing the underlying elliptic curve and, consequently, without the computational requirements inherent to the set up of elliptic curves at random. Also, we expose the Diffie-Hellman key agreement protocol with this system acting as the underlying mathematical problem.

Applying quick exponentiation for block upper triangular matrices

Abstract The best method for exponentiation is highly dependant on the algebraic set employed. Block upper triangular matrices defined in Z p , have very interesting properties for multiple applications, in which exponentiation is very important in order to achieve adequate performance. We analyze the usage of quick exponentiation methods with these matrices and, as a practical application, we propose a new public-key cryptosystem and digital signature scheme based on a generalization of the well known discrete logarithm problem to block upper triangular matrices.

New highlights and a new centrality measure based on the Adapted PageRank Algorithm for urban networks

A centrality measure, with a practical application in urban network, has been proposed.The measure establishes a ranking of nodes focused on the topological data distribution.The amount of data associated with the nodes of the network has a great importance.The topology of the network and the information associated to the nodes are involved. The Adapted PageRank Algorithm (APA) proposed by Agryzkov etźal. 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.

Measuring urban activities using Foursquare data and network analysis: a case study of Murcia Spain

ABSTRACT Among social networks, Foursquare is a useful reference for identifying recommendations about local stores, restaurants, malls or other activities in the city. In this article, we consider the question of whether there is a relationship between the data provided by Foursquare regarding users’ tastes and preferences and fieldwork carried out in cities, especially those connected with business and leisure. Murcia was chosen for case study for two reasons: its particular characteristics and the prior knowledge resulting from the fieldwork. Since users of this network establish, what may be called, a ranking of places through their recommendations, we can plot these data with the objective of displaying the characteristics and peculiarities of the network in this city. Fieldwork from the city itself gives us a set of facilities and services observed in the city, which is a physical reality. An analysis of these data using a model based on a network centrality algorithm establishes a classification or ranking of the nodes that form the urban network. We compare the data extracted from the social network with the data collected from the fieldwork, in order to establish the appropriateness in terms of understanding the activity that takes place in this city. Moreover, this comparison allows us to draw conclusions about the degree of similarity between the preferences of Foursquare users and what was obtained through the fieldwork in the city.

Analyzing the commercial activities of a street network by ranking their nodes: a case study in Murcia, Spain

Urban researchers and planners are often interested in understanding how economic activities are distributed in urban regions, what forces influence their special pattern and how urban structure and functions are mutually dependent. In this paper, we want to show how an algorithm for ranking the nodes in a network can be used to understand and visualize certain commercial activities of a city. The first part of the method consists of collecting real information about different types of commercial activities at each location in the urban network of the city of Murcia, Spain. Four clearly differentiated commercial activities are studied, such as restaurants and bars, shops, banks and supermarkets or department stores, but obviously we can study other. The information collected is then quantified by means of a data matrix, which is used as the basis for the implementation of a PageRank algorithm which produces a ranking of all the nodes in the network, according to their significance within it. Finally, we visualize the resulting classification using a colour scale that helps us to represent the business network.

An application of a self-organizing model to the design of urban transport networks

A 2D triangle mesh simplification model is described in this paper, with the main objective of preserving the shape of the original mesh. The proposed model consists of a self-organizing algorithm whose objective is to generate the positions of the nodes of the simplified mesh; afterwards, a triangulation algorithm is performed to reconstruct the triangles of the new simplified mesh. The self-organizing algorithm is an unsupervised learning algorithm that provides a set of nodes representing the best approximation of the original mesh. An adaptation of the neural network algorithm is proposed with the primary objective to work in the context of urban transport networks. We verify the effectiveness of this model through the design and development of some urban network problems. Specifically, the algorithm is applied to two real problems, the first one is the design of a tramway network in a town, and the second one is the design of an information point network within a real bus transport network.

A model to simplify 2D triangle meshes with irregular shapes

We present a 2D triangle mesh simplification model which is able to produce high quality approximations of any original planar mesh, regardless of the shape of the original mesh. This method consists of two phases: a self-organizing algorithm and a triangulation algorithm. The self-organizing algorithm is an unsupervised incremental clustering algorithm which provides us a set of nodes representing the best approximation of the original mesh. The triangulation algorithm reconstructs the simplified mesh from the planar points obtained by the self-organizing training process. Some examples are detailed with the purpose of demonstrating the ability of the model to perform the task of simplifying an original mesh with irregular shape.

A Model to Visualize Information in a Complex Streets' Network

This paper discusses a process to graphically view and analyze information obtained from a network of urban streets, using an algorithm that establishes a ranking of importance of the nodes of the network itself. The basis of this process is to quantify the network information obtained by assigning numerical values to each node, representing numerically the information. These values are used to construct a data matrix that allows us to apply a classification algorithm of nodes in a network in order of importance. From this numerical ranking of the nodes, the process finish with the graphical visualization of the network. An example is shown to illustrate the whole process.