Revealing differences between passenger demand and bus service networks: a mixed membership community detection approach

Public transit networks are critical to urban systems, yet aligning fixed bus services with dynamic and spatially heterogeneous passenger demand remains challenging. This study develops a network-based framework to compare passenger demand and bus service from a community-structure perspective. Both networks are formulated within a directed weighted degree-corrected mixed-membership model. Community memberships are estimated through co-clustering based on mixed simplex and cone geometry, and a directed weighted overlapping modularity determines a common community resolution. Applying the framework to Nanjing’s bus network reveals organisational differences, directional asymmetry, and station-level alignment. Despite broadly similar spatial distributions, demand communities are relatively continuous, block-like, and polycentric, whereas bus service communities are more concentrated and corridor-oriented, reflecting route organisation and infrastructure constraints. These findings provide a basis for understanding how passenger demand and bus service differ in their network organisation, with implications for bus network planning and service adjustment.