Hélder Cerqueira Alves

Abstract
In this paper we introduce and develop the concept of interval-weighted networks (IWN), a novel approach in Social Network Analysis, where the edge weights are represented by closed intervals composed with precise information, comprehending intrinsic variability. We extend IWN for both Newman's modularity and modularity gain and the Louvain algorithm, considering a tabular representation of networks by contingency tables. We apply our methodology to two real-world IWN. The first is a commuter network in mainland Portugal, between the twenty three NUTS 3 Regions (IWCN). The second focuses on annual merchandise trade between 28 European countries, from 2003 to 2015 (IWTN). The optimal partition of geographic locations (regions or countries) is developed and compared using two new different approaches, designated as Classic Louvain and Hybrid Louvain , which allow taking into account the variability observed in the original network, thereby minimizing the loss of information present in the raw data. Our findings suggest the division of the twenty three Portuguese regions in three main communities for the IWCN and between two to three country communities for the IWTN. However, we find different geographical partitions according to the community detection methodology used. This analysis can be useful in many real-world applications, since it takes into account that the weights may vary within the ranges, rather than being constant.

Abstract
Centrality measures are used in network science to assess the centrality of vertices or the position they occupy in a network. There are a large number of centrality measures according to some criterion. However, the generalizations of the most well-known centrality measures for weighted networks, degree centrality, closeness centrality and betweenness centrality have solely assumed the edge weights to be constants. This article proposes a methodology to generalize degree, closeness and betweenness centralities taking into account the variability of edge weights in the form of closed intervals (interval-weighted networks, IWN). We apply our centrality measures approach to two real-world IWN. The first is a commuter network in mainland Portugal, between the 23 NUTS 3 Regions. The second focuses on annual merchandise trade between 28 European countries, from 2003 to 2015.