Measuring the importance of nodes in a network with a centrality measure is an core task in any network application. There many measures available and it is speculated that many encode similar information. We give an explicit non-linear relationship between two of the most popular measures of node centrality: degree and closeness. Based on a shortest-path tree approximation, we give an analytic derivation that shows the inverse of closeness is linearly dependent on the logarithm of degree. We show that our hypothesis works well for a range of networks produced from stochastic network models and for networks derived from 130 real-world data sets. We connect our results with previous results for other network distance scales such as average distance. Our results imply that measuring closeness is broadly redundant unless our relationship is used to remove the dependence on degree from closeness. The success of our relationship suggests that most networks can be approximated by shortest-path spanning trees which are all statistically similar two or more steps away from their root nodes.
Cities are typical dynamic complex systems that connect people and facilitate interactions. Revealing general collective patterns behind spatiotemporal interactions between residents is crucial for various urban studies, of which we are still lacking a comprehensive understanding. Massive cellphone data enable us to construct interaction networks based on spatiotemporal co-occurrence of individuals. The rank-size distributions of dynamic population of locations in all unit time windows are stable, although people are almost constantly moving in cities and hot-spots that attract people are changing over time in a day. A larger city is of a stronger heterogeneity as indicated by a larger scaling exponent. After aggregating spatiotemporal interaction networks over consecutive time windows, we reveal a switching behavior of cities between two states. During the “active” state, the whole city is concentrated in fewer larger communities, while in the “inactive” state, people are scattered in smaller communities. Above discoveries are universal over three cities across continents. In addition, a city stays in an active state for a longer time when its population grows larger. Spatiotemporal interaction segregation can be well approximated by residential patterns only in smaller cities. In addition, we propose a temporal-population-weighted-opportunity model by integrating a time-dependent departure probability to make dynamic predictions on human mobility, which can reasonably well explain the observed patterns of spatiotemporal interactions in cities.
The interactions between vulnerability and human activities have largely been regarded in terms of the level of risk they pose, both internally and externally, for certain groups of disadvantaged individuals and regions/areas. However, to date, very few studies have attempted to develop a comprehensive composite regional vulnerability index, in relation to travel, housing, and social deprivation, which can be used to measure vulnerability at an aggregated level in the social sciences. Therefore, this research aims to develop a composite regional vulnerability index with which to examine the combined issues of travel, housing and socio-economic vulnerability (THASV index). It also explores the index’s relationship with the impacts of the COVID-19 pandemic, reflecting both social and spatial inequality, using Greater London as a case study, with data analysed at the level of Middle Layer Super Output Areas (MSOAs). The findings show that most of the areas with high levels of composite vulnerability are distributed in Outer London, particularly in suburban areas. In addition, it is also found that there is a spatial correlation between the THASV index and the risk of COVID-19 deaths, which further exacerbates the potential implications of social deprivation and spatial inequality. Moreover, the results of the multiscale geographically weighted regression (MGWR) show that the travel and socio-economic indicators in a neighbouring district and the related vulnerability indices are strongly associated with the risk of dying from COVID-19. In terms of policy implications, the findings can be used to inform sustainable city planning and urban development strategies designed to resolve urban socio-spatial inequalities and the potential related impacts of COVID-19, as well as guiding future policy evaluation of urban structural patterns in relation to vulnerable areas.
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