Learning about the liveability of cities from young migrants using the combinatiorial Hodge theory approach

In declining and ageing societies, local communities face the `risk of eventual extinction.' In Japan, a population equivalent to that of an entire city is lost every year, representing one of the most severe cases of population decline. Thus, attracting young people has become a policy priority for many local municipalities in Japan, prompting the implementation of numerous initiatives to improve the liveability of affected cities. However, what exactly is a liveable city? To determine this, a concrete measure of liveability is required to serve as a key performance indicator (KPI) for local governments to adopt. In this study, we propose empirical liveability based on people's votes with their feet, following Tiebout's argument (Tiebout,1956), and derive that such liveability can be quantified using the `potential' in the combinatorial Hodge theory, directly calculated from migration data only. As a case study, we measure the empirical liveability of municipalities in Japan for specific populations -- families with small children and women of reproductive age. Then, using the empirical liveability as dependent variables, we perform a regression analysis to identify factors related to liveability. This method is applicable to various datasets on migration, categorized by ethnicity, education, skill level, income and other attributes, and provides valuable statistics for urban planning and policymaking.