The Topology of Hardship: Empirical Curriculum Graphs and Structural Bottlenecks in Engineering Degrees

Engineering degrees are often perceived as "hard", yet this hardness is usually discussed in terms of content difficulty or student weaknesses rather than as a structural property of the curriculum itself. Recent work on course-prerequisite networks and curriculum graphs has shown that study plans can be modelled as complex networks with identifiable hubs and bottlenecks, but most studies rely on official syllabi rather than on how students actually progress through the system (Simon de Blas et al., 2021; Stavrinides & Zuev, 2023; Yang et al., 2024; Wang et al., 2025). This paper introduces the notion of topology of hardship: a quantitative description of curriculum complexity derived from empirical student trajectories in long-cycle engineering programmes. Building on the CAPIRE framework for multilevel trajectory modelling (Paz, 2025a, 2025b), we reconstruct degree-curriculum graphs from enrolment and completion data for 29 engineering curricula across several cohorts. For each graph we compute structural metrics (e.g., density, longest path, bottleneck centrality) and empirical hardship measures capturing blocking probability and time-to-progress. These are combined into a composite hardship index, which is then related to observed dropout rates and time to degree. Our findings show that curriculum hardness is not a vague perception but a measurable topological property: a small number of structurally dense, bottleneck-heavy curricula account for a disproportionate share of dropout and temporal desynchronisation. We discuss implications for curriculum reform, accreditation, and data-informed policy design.