Selection of Centrality Measures Using Self-Consistency and Bridge Axioms

We consider several families of network centrality measures induced by graph kernels, which include some well-known measures and many new ones. The Self-consistency and Bridge axioms, which appeared earlier in the literature, are closely related to certain kernels and one of the families. We obtain a necessary and sufficient condition for Self-consistency, a sufficient condition for the Bridge axiom, indicate specific measures that satisfy these axioms, and show that under some additional conditions they are incompatible. PageRank centrality applied to undirected networks violates most conditions under study and has a property that according to some authors is ``hard to imagine'' for a centrality measure. We explain this phenomenon. Adopting the Self-consistency or Bridge axiom leads to a drastic reduction in survey time in the culling method designed to select the most appropriate centrality measures.Comment: 23 pages, 5 figures. A reworked versio

Distance-based accessibility indices

The paper attempts to develop a suitable accessibility index for networks where each link has a value such that a smaller number is preferred like distance, cost, or travel time. A measure called distance sum is characterized by three independent properties: anonymity, an appropriately chosen independence axiom, and dominance preservation, which requires that a node not far to any other is at least as accessible. We argue for the need of eliminating the independence property in certain applications. Therefore generalized distance sum, a family of accessibility indices, will be suggested. It is linear, considers the accessibility of vertices besides their distances and depends on a parameter in order to control its deviation from distance sum. Generalized distance sum is anonymous and satisfies dominance preservation if its parameter meets a sufficient condition. Two detailed examples demonstrate its ability to reflect the vulnerability of accessibility to link disruptions

How to choose the most appropriate centrality measure?

We propose a new method to select the most appropriate network centrality measure based on the user's opinion on how such a measure should work on a set of simple graphs. The method consists in: (1) forming a set $\cal F$ of candidate measures; (2) generating a sequence of sufficiently simple graphs that distinguish all measures in $\cal F$ on some pairs of nodes; (3) compiling a survey with questions on comparing the centrality of test nodes; (4) completing this survey, which provides a centrality measure consistent with all user responses. The developed algorithms make it possible to implement this approach for any finite set $\cal F$ of measures. This paper presents its realization for a set of 40 centrality measures. The proposed method called culling can be used for rapid analysis or combined with a normative approach by compiling a survey on the subset of measures that satisfy certain normative conditions (axioms). In the present study, the latter was done for the subsets determined by the Self-consistency or Bridge axioms.Comment: 26 pages, 1 table, 1 algorithm, 8 figure

Multidimensional Minimal Spanning Tree: The Bursa Malaysia

The stock market has constituted a complex system since the interrelationships among the stocks are complicated and unpredictable. Moreover, the stock price does not stagnate at a certain price all the time but the price keeps changing from minute to minute during the transaction hours. Thus, it is quite difficult to indicate which stock influences the performances of other stocks as well as the behaviours of the stocks in a network. The economic information might be misleading and incomplete if the analysis applies with univariate time series of stock price only as each stock is represented by four features of the price. To obtain the complete information of the Bursa Malaysia stock network as well as the interrelationships among the stocks, multivariate time series of stocks are measured by using RV coefficient. Besides, minimum spanning tree and centrality measures are applied in this paper in order to construct the stock network virtually and determine the behaviours of the stocks by using the recent data of top 100 stocks in Bursa Malaysia

Measuring centrality by a generalization of degree

Network analysis has emerged as a key technique in communication studies, economics, geography, history and sociology, among others. A fundamental issue is how to identify key nodes in a network, for which purpose a number of centrality measures have been developed. This paper proposes a new parametric family of centrality measures called generalized degree. It is based on the idea that a relationship to a more interconnected node contributes to centrality in a greater extent than a connection to a less central one. Generalized degree improves on degree by redistributing its sum over the network with the consideration of the global structure. Application of the measure is supported by a set of basic properties. A sufficient condition is given for generalized degree to be rank monotonic, excluding counter-intuitive changes in the centrality ranking after certain modifications of the network. The measure has a graph interpretation and can be calculated iteratively. Generalized degree is recommended to apply besides degree since it preserves most favorable attributes of degree, but better reflects the role of the nodes in the network and has an increased ability to distinguish between their importance

Rank monotonicity in centrality measures

A measure of centrality is rank monotone if after adding an arc x -> y, all nodes with a score smaller than (or equal to) y have still a score smaller than (or equal to) y. If, in particular, all nodes with a score smaller than or equal to y get a score smaller than y (i.e., all ties with y are broken in favor of y), the measure is called strictly rank monotone. We prove that harmonic centrality is strictly rank monotone, whereas closeness is just rank monotone on strongly connected graphs, and that some other measures, including betweenness, are not rank monotone at all (sometimes not even on strongly connected graphs). Among spectral measures, damped scores such as Katz's index and PageRank are strictly rank monotone on all graphs, whereas the dominant eigenvector is strictly monotone on strongly connected graphs only

Mobile app recommendations using deep learning and big data

Dissertation presented as the partial requirement for obtaining a Master's degree in Statistics and Information Management, specialization in Marketing Research e CRMRecommender systems were first introduced to solve information overload problems in enterprises. Over the last decades, recommender systems have found applications in several major websites related to e-commerce, music and video streaming, travel and movie sites, social media and mobile app stores. Several methods have been proposed over the years to build recommender systems. The most popular approaches are based on collaborative filtering techniques, which leverage the similarities between consumer tastes. But the current state of the art in recommender systems is deep-learning methods, which can leverage not only item consumption data but also content, context, and user attributes. Mobile app stores generate data with Big Data properties from app consumption data, behavioral, geographic, demographic, social network and user-generated content data, which includes reviews, comments and search queries. In this dissertation, we propose a deep-learning architecture for recommender systems in mobile app stores that leverage most of these data sources. We analyze three issues related to the impact of the data sources, the impact of embedding layer pretraining and the efficiency of using Kernel methods to improve app scoring at a Big Data scale. An experiment is conducted on a Portuguese Android app store. Results suggest that models can be improved by combining structured and unstructured data. The results also suggest that embedding layer pretraining is essential to obtain good results. Some evidence is provided showing that Kernel-based methods might not be efficient when deployed in Big Data contexts