Takaaki Aoki

Modern society relies on complex supply chains to sustain the flow of goods and services that are essential to daily life. While traditional supply chain theory assumes a clear, hierarchical flow from upstream suppliers to downstream customers, observable real-world transaction networks rarely exhibit this acyclic structure. Instead, detailed inter-firm data reveal that interwoven networks are heavily entangled by cyclic flows. Consequently, without appropriate partitioning of these massive inter-firm networks, the latent flow-hierarchical structures that are central to supply chain concepts remain obscure. To address this analytical challenge, we introduce the flow-Hierarchical Community Network Extraction (f-HiCoNE) framework. By applying combinatorial Hodge decomposition, this approach disentangles the complex inter-firm network by isolating the acyclic gradient flow to quantify the flow-hierarchical parts and partition the graph. By applying f-HiCoNE to a nationwide transaction dataset of approximately 650,000 firms, we extracted clusters exhibiting the flow-hierarchical organisation that supply-chain theory assumes. These clusters demonstrated strong flow-hierarchical organisation, wherein the upstream-downstream positioning of firms was well described by local scalar potentials, revealing distinct geographically localised industrial ecosystems. This study provides a map that could help firms understand their surrounding environment and locate their position within an inter-firm network and opens a new research avenue focused on flow-hierarchy clustering in supply chain analysis.

Understanding human mobility is vital to solving societal challenges, such as epidemic control and urban transportation optimization. Recent advancements in data collection now enable the exploration of dynamic mobility patterns in human flow. However, the vast volume and complexity of mobility data make it difficult to interpret spatiotemporal patterns directly, necessitating effective information reduction. The core challenge is to balance data simplification with information preservation: methods must retain location-specific information about human flows from origins to destinations while reducing the data to a comprehensible level. This study proposes a two-step dimensionality reduction framework: First, combinatorial Hodge theory is applied to the given origin–destination (OD) matrices with timestamps to construct a set of potential landscapes of human flow, preserving imbalanced trip information between locations. Second, principal component analysis (PCA) expresses the time series of potential landscapes as a linear combination of a few static spatial components, with their coefficients representing temporal variations. The framework systematically decouples the spatial and temporal components of the given data. By implementing this two-step reduction method, we reveal large weight variations during a pandemic, characterized by an overall decline in mobility and stark contrasts between weekdays and holidays. These findings demonstrate the effectiveness of our framework in uncovering complex mobility patterns and its potential to inform urban planning and public health interventions.

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 ¹ , 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.

Human mobility, encompassing activities ranging from daily commutes to long-term migration, plays a critical role in shaping societal structures, economic activities, and cultural exchanges. Its complex and multifaceted nature, driven by factors such as socioeconomic conditions, technological advancements, policy interventions, and environmental changes, has made human mobility a central focus of interdisciplinary research across various domains. This review explores recent advancements in human mobility studies, focusing on the integration of diverse data sources to analyze movement patterns. Key models of human mobility are introduced, addressing both individual-level dynamics, such as individual travel behavior, and population-level phenomena, such as migration flows. Furthermore, this review summarizes the practical applications of human mobility, particularly in urban planning and epidemic modeling, highlighting the significant implications of a comprehensive understanding of human mobility. Understanding these dynamics is essential for developing data-driven strategies to enhance urban infrastructure, manage public health risks, optimize transportation networks, and respond effectively to global mobility trends.

Migration is a major decision to leave one place and move to another, involving career and life changes. Migration flows provide people's comparisons across places about which is better to live by ''vote with their feet'' (Tiebout, 1956). However, is it possible to derive a consistent measure of the liveability of cities from these flows? We propose a combinatorial-Hodge-theory approach: the empirical liveability of cities is evaluated by a potential governing unbalanced, acyclic migrations between cities. As a case study, we measure the liveability of municipalities in Japan for specific populations such as families with small children and women of reproductive age in a population-decline society. Using these potentials as dependent variables, we perform a regression analysis to identify the factors relevant to liveability. We also derive analytical expressions that allow us to interpret as potentials the standards of living or utilities, estimated in the economics literature (Douglas & Wall, 1993; Douglas, 1997; Douglas & Wall, 2000; Wall, 2001; Nakajima & Tabuchi, 2011). The proposed method extracts a consistent metric of interval scale from the non-transitive, pairwise comparison between locations and provides substantial statistics for urban planning by policymakers.

The availability of increasingly abundant mobility data in recent years has opened up new avenues for researchers to unravel human mobility patterns. Data aggregation methods have been introduced to gain a quantitative understanding of collective individual movements using these data. Nevertheless, the widely adopted origin–destination (OD) aggregation method for human mobility data lacks an essential piece of information: home location, which plays a vital role in characterizing individual movement patterns. In this study, we propose a novel data aggregation approach called home–origin–destination (HOD) with the aim of improving the accuracy of human mobility estimation. We compare the performance of various data aggregation methods for estimating population distribution. Our experimental results reveal more realistic mobility patterns when incorporating estimated home information, where individuals move out in the morning and return home before midnight. To further evaluate the effectiveness of the HOD approach, we conduct an entropy analysis to measure the unpredictability of human mobility. The HOD results exhibit lower entropy values than those in the other two cases, OD and home–destination (HD). These findings underscore the importance of incorporating home information in understanding and modeling human mobility. By leveraging the HOD data aggregation method, we can achieve more accurate population distribution estimates and capture the inherent dynamics of human movement

Human flow data are rich behavioral data relevant to people’s decision-making regarding where to live, work, go shopping, etc., and provide vital information for identifying city centers. However, it is not as easy to understand massive relational data, and datasets have often been reduced merely to the statistics of trip counts at destinations, discarding relational information from origin to destination. In this study, we propose an alternative center identification method based on human mobility data. This method extracts the scalar potential field of human trips based on combinatorial Hodge theory. It detects not only statistically significant attractive locations as the sinks of human trips but also significant origins as the sources of trips. As a case study, we identify the sinks and sources of commuting and shopping trips in the Tokyo metropolitan area. This aim-specific analysis leads to a combinatorial classification of city centers based on the distinct aspects of human mobility. The proposed method can be applied to other mobility datasets with relevant properties and helps us examine the complex spatial structures in contemporary metropolitan areas from the multiple perspectives of human mobility.

Human flow in cities indicates social activity and can reveal urban spatial structures based on human behaviours for relevant applications. Scalar potential is a mathematical concept that, when properly applied, can provide an intuitive view of human flow. However, the definition of such a potential in terms of the origin-destination flow matrix and its feasibility remain unresolved. In this case, we use Hodge–Kodaira decomposition, which uniquely decomposes a matrix into a potential-driven (gradient) flow and a curl flow. We depict the potential landscapes in cities resulting from commuting flow and reveal how the landscapes have either changed or remained unchanged by years or methods of transportation. We then determine how well the commuting flow is described by the potential, by evaluating the percentage of the gradient component for metropolitan areas in the USA and show that the gradient component is almost 100% in several areas; in other areas, however, the curl component is dominant, indicating the importance of circular flow along with triangles of places. The potential landscape provides an easy-to-use visualisation tool for showing the attractive places of human flow and will help in a variety of applications such as commerce, urban design, and epidemic spreading.

Emergence of cities and road networks have characterised human activity and movement over millennia. However, this anthropogenic infrastructure does not develop in isolation, but is deeply embedded in the natural landscape, which strongly influences the resultant spatial patterns. Nevertheless, the precise impact that landscape has on the location, size and connectivity of cities is a long-standing, unresolved problem. To address this issue, we incorporate high-resolution topographic maps into a Turing-like pattern forming system, in which local reinforcement rules result in co-evolving centres of population and transport networks. Using Italy as a case study, we show that the model constrained solely by topography results in an emergent spatial pattern that is consistent with Zipf’s Law and comparable to the census data. Thus, we infer the natural landscape may play a dominant role in establishing the baseline macro-scale population pattern, that is then modified by higher-level historical, socio-economic or cultural factors.