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

TL;DR

The study tackles the problem of defining a concrete KPI for city liveability in the context of severe population decline in Japan. It introduces an empirical liveability score derived from migration flows by applying the combinatorial Hodge theory to the origin–destination matrix, extracting a per-location potential $s_i$ that reflects liveability differences. By decomposing migration on a network into gradient, curl, and harmonic components, the authors show that the gradient part yields a globally consistent, interval-scale liveability measure that extends traditional net-migration and regional-utility concepts, while incorporating distance through edge weights. The method is demonstrated with WoRA and FwSC populations across Japanese municipalities, revealing distinct spatial patterns and enabling regression analyses that identify group-specific determinants of liveability, with implications for targeted urban policy. The approach is data-driven, generalizable to other migration datasets, and provides a practical framework for evidence-based place-based policymaking.

Abstract

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.

Figures (9)

• Figure 1: Hodge decomposition of a migration flow. The network in the left panel shows migration data among six locations. The black numbers of links denote the net movements between them. This flow is uniquely and orthogonally decomposed into gradient, harmonic, and curl flows in the combinatorial Hodge theory (top panel). Blue numbers in the gradient flow denote the potentials whose regional differences give the global acyclic part of flows and quantify the empirical liveability of each location. The other two flows are perpendicular to the acyclic flow: these flows are circular among locations, and incoming and outgoing fluxes are balanced.
• Figure 2: Percentage of origin--destination (OD) pairs $(i,j)$ with nonzero migration ($M_{ij} > 0$ or $M_{ji} > 0$) as a function of road distance $d_{ij}$ in the Japanese migration OD matrix, $M$. (A) The full range of road distance. (B) A focused range of $[0,200]$ km.
• Figure 3: (A,B) The potential landscape of migration flows for women of reproductive age in the entire target area in Japan (A) and the focused area around Tokyo (B), shown in a green rectangle in the left panel. The unit of the potential analysis is municipality in Japan. The black boundary illustrates the area of the special wards of Tokyo, known as the central area of the metropolis. The green boundary indicates Tokyo Metropolitan Employment Area as an alternative to an official metropolitan area. (C,D) Same as A, B, but for families with small children. Larger images of Panels A and C are in Supplementary Information.
• Figure 4: Comparison between the women of reproductive age and families with small children groups by their potentials. A point corresponds to a municipality.
• Figure 5: (A) Combinatorial classification by statistical testing for the migrations of two specific populations: women of reproductive age (WoRA) and families with small children (FwSC). (B) The map of the classified municipalities in the entire target area in Japan. Significant municipalities of migrations are indicated by colours, according to the classification scheme shown in A. (C) A focused map of a Tokyo area, indicated by a rectangle in panel (B). The black boundary illustrates the area of the special wards of Tokyo, known as the central area of the metropolis. The green boundary indicates Tokyo Metropolitan Employment Area as an alternative to an official metropolitan area. (D) A focused map of a Kyoto-Osaka-Kobe (Keihanshin) area, indicated by a rectangle in panel (B). The black boundary illustrates the area of Osaka city as the central area, and the green boundary indicates Osaka Metropolitan Employment.