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
The demographic aging presents significant intervention challenges to institutions within the elderly care management system, such as the National Republican Guard (GNR) with its program “65 years—elderly in safety,” which aims to minimize the effects of social isolation and the experience of loneliness. This exploratory study had as its main goal to understand how elderly individuals aged 65 and over who participate in this GNR program experience loneliness and to identify the social and emotional challenges that arise from it, particularly regarding mental health. Twenty elderly people were surveyed, out of a total of 40, and identified and monitored by the GNR. This is a non-probabilistic quota sample, selected according to gender and age group. In addition to a sociodemographic characterization survey, the UCLA Loneliness Scale, the SELSA-S (Social and Emotional Loneliness Scale for Adults), and the Geriatric Depression Scale (GDS 15) were used. The surveyed elderly present a moderate level of loneliness, with most having family but not finding in it the necessary social support to meet their needs and promote their mental health and well-being. The fact that they live in rural areas, far from large population centers, creates barriers to fulfilling their various needs and often, as evidenced by the data collected, leads them to develop depressive symptoms. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

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.