Curricula are more than a checklist of courses. They are living maps of how a university whispers from one topic to the next, from introductory ideas to the frontier edges of a discipline. A pair of researchers from Caltech and Georgia Tech have given us a new vocabulary for reading that map, revealing its shape at a macro scale. The study, led by Konstantin Zuev and Pavlos Stavrinides, translates course catalogs into a language of networks and then asks: what does the whole landscape look like when you zoom out far enough to see breadth, depth, and flow with the same clarity you reserve for traffic on a highway or the spread of ideas across a field? The work, which analyzes real curricula from three universities and two simple random models, is a concerted push toward comparing entire course-prerequisite networks rather than just the importance of individual courses. The data and results come from a collaboration that spans Caltech and Georgia Tech, with real-world examples drawn from the Cyprus University of Technology, Caltech, and Johns Hopkins University. The authors hope these metrics will help universities design smarter programs and advisors guide students through the evolving architecture of knowledge.
Key idea that unlocks this work is simple in spirit but powerful in consequence: once you strip away redundant links that do not alter what you can reach in a curriculum, the essential shape of a program remains. By focusing on the macro structure that survives this pruning, the researchers define a topological stratification that slices courses into layers from introductory to advanced. With that stratified view in hand, they define three global measures that summarize an entire curriculum: breadth, depth, and flux. These are not just mathematical toys; they are new levers for comparing curricula across institutions and for asking what a curriculum emphasizes about breadth of topics, the depth of the learning path, and the flow of knowledge from one layer to the next.
A New Lens on Course Prerequisites
The core object of study is a course-prerequisite network, a directed acyclic graph in which nodes are courses and links point from prerequisites to the courses that depend on them. If you study differential geometry, for example, that course typically lies before general relativity in the scholarly chain. But real curricula are messy. There are often multiple paths to the same advanced topic, and some links are redundant in the sense that their effect is already captured by other prerequisites along a longer route. The team treats such redundancy with a mathematical operation called transitive reduction, which keeps every course but trims away the extraneous arcs that do not change reachability. This pruning is not just cosmetic. It guarantees that any global measure should be invariant to these simplifications, ensuring that the macro properties reflect genuine structure rather than accidental clutter.
Intuition helps here: imagine a city map where some roads are duplicates of longer routes in disguise. If you want to understand the flow of traffic between neighborhoods, you can remove the duplicate lanes and still capture how people actually travel. That is the idea behind transitive reduction in a curriculum network. The authors show that once you reduce the network, the global measures they introduce remain the same. This invariance is crucial because it lets researchers compare curricula across institutions and datasets without worrying that tiny, potentially arbitrary link patterns will distort the big picture.
Two concepts sit at the heart of the approach to topology. The first is the recognition that real course networks are DAGs but not necessarily ordered in a single canonical way. In strictly ordered graphs, you can line up nodes in a fixed sequence and read off a simple progression from one layer to the next. In a real university catalog you cannot impose a single natural order that captures every prerequisite chain. The second idea is topological stratification, a principled way to partition courses into strata S1, S2, …, ST where every course in Stratum t has its prerequisites drawn from Strata strictly below t, and its postrequisites lie in higher strata. Within a stratum, courses are not connected by prerequisites, so their ordering is effectively irrelevant for the macro measures. This layered view gives us a clean way to talk about overall curriculum structure without getting lost in local detours.
Takeaway from this section: transitive reduction and topological stratification fuse to give a robust, multi-layered picture of how a curriculum builds knowledge up the ladder, layer by layer, without overcounting paths that are already implied by longer routes. This layered view is the canvas on which breadth, depth, and flux are painted.
Breadth, Depth, and Flux: Macro View of Curricula
The three global measures are designed to capture aspects of a curriculum that usually hide behind micro analysis of which course is central or which department dominates a program. Breadth, depth, and flux together form a trio that describes how wide the curriculum reaches, how deep the knowledge ladder runs, and how information percolates through the layers of courses.
Breadth is defined as the average size of a layer. If you tally how many courses sit in S1, S2, …, ST and take the average, you get a single number that reflects how much is packed into each layer on average. A larger breadth means the curriculum spreads across more courses in each layer, signaling a broad but perhaps shallower education. A more compact stratification with few but large layers points to a broader, more horizontal spread of topics anchored in every step of the ladder. Importantly, breadth is invariant to transitive reduction, so pruning redundant links leaves this macro property unchanged. The number of strata T is also interpretable as the length of the longest directed path through the curriculum; breadth and depth thus trade off against each other in a neat, measurable way.
Depth aims to quantify how far the knowledge can be pushed when you travel from the most introductory layers to the very end. It is defined by the average depth of the set of zero-out-degree nodes, the courses at the far frontier of the curriculum. In simple terms, depth gauges how long the culminating courses sit from the starting blocks. If you were to pick a final course at random from the set of deepest courses and trace back through its prerequisites to the first course, depth would reflect the length of that journey. The probabilistic interpretation is elegant: the depth is the expected layer index of a randomly chosen terminal course, which ties directly to the idea of a knowledge journey with a typical number of steps from start to finish.
Flux is perhaps the trickiest of the trio, a measure of how knowledge flows across neighboring layers. It looks at the net number of links connecting a stratum to the next one versus the previous one, normalized by the layer size. If a layer tends to feed forward more than it receives from the layer before, it has positive flux; if it tends to pull knowledge back or stall, the flux is negative. Flows tend to decline as you climb to higher strata because introductory material more often acts as a springboard for deeper topics, while advanced courses rely on earlier knowledge but offer fewer direct prerequisites back to the start. Flux captures this asymmetry in a single, interpretable number per stratum, and then averages across all strata to give a curriculum-wide sense of how knowledge moves through the system.
When you plot breadth against depth for various curricula, a clear—and perhaps intuitive—pattern emerges: broader curricula tend to be shallower, and deeper curricula tend to be narrower. The network view makes this tradeoff concrete rather than anecdotal. The article shows, for the three real curricula examined, how breadth and depth align with the size and structure of the program, and it demonstrates how flux behaves across strata as knowledge travels upward through the ladder. This macro view is the scorecard of a curriculum, filling in gaps that micro measures might miss.
Why this matters is not merely academic. If universities are trying to balance offering a wide array of courses with building a coherent, progressive path for students, breadth and depth provide quantifiable targets. Flux offers a diagnostic for where a program leaks knowledge forward or stalls backward, signaling potential bottlenecks or misalignments between introductory tracks and advanced offerings. The trio thus becomes a practical toolkit for curriculum reform and advising, not just a theoretical curiosity.
Real Curricula Meet Random Graphs
To see whether these new measures capture something fundamental about curricula, the authors turned to three distinct real networks: the Cyprus University of Technology, the California Institute of Technology, and Johns Hopkins University. They also created two classes of synthetic networks that mimic certain statistical properties of real curricula. One class mimics the classic Erdős–Rényi approach, generating random DAGs with a fixed number of nodes and edges. The other class uses a Karrer-Newman style configuration model for directed acyclic graphs, which preserves degree sequences and, crucially, imposes a topological ordering that makes the synthetic networks more structurally similar to real curricula in certain respects.
What surprised the researchers was the mixed success of these models. The simple Erdős–Rényi CPNs deviate substantially from real curricula on all three macro measures, underscoring that random edge placement cannot emulate the deliberate architecture of education. The KN model, which respects the degree sequence and uses a more structured approach to linking prerequisites, fares better for some networks. In particular, it captures the breadth and depth of the smallest real curriculum, CUT, quite well on average. But for larger and more diverse programs like CIT and JHU, the KN model struggles to reproduce the observed breadth and depth. The implication is not that KN or ER are useless, but that real curricula embody a level of organizational nuance that these simple generative schemes miss, especially when the network is large and heterogeneous in its offerings.
The results also illuminate how flux behaves in practice. In the KN-generated samples, depth and flux align most closely with the CUT pattern, while CIT and JHU reveal a gap between what synthetic models can reproduce and what the real networks exhibit. The flux values across strata in the real curricula tend to decline as you move toward higher strata, in line with the story that introductory material tends to propel students forward, while advanced topics pull knowledge forward but rely on a longer chain of prerequisites. This subtle curvature in flux across strata provides a window into how a curriculum channels or slows the movement of knowledge along the ladder of attainment.
From a broader perspective, the study shows that macro-scale measures can serve as a litmus test for generative models. If a model can reproduce breadth, depth, and flux in a way that tracks real curricula, it is offering a plausible story for how courses are conceived and linked. If not, it signals that the model is overlooking some essential mechanism, whether it is how departments coordinate, how advising shapes course choices, or how evolving disciplinary frontiers reshape prerequisites over time. The authors emphasize that there is no single authority coordinating every course in a modern university; decisions are decentralized and local. Yet the global patterns emerge, suggesting that macro-level laws or constraints might govern curriculum evolution even without a central planner.
Practical implications flow from this. If a university can map its own CPN and compute breadth, depth, and flux, it can compare itself to peer institutions in a meaningful way that goes beyond ranking just research output or faculty count. It can identify strata where more content could be added to improve breadth without sacrificing depth, or where flux is unbalanced and could be nudged to create smoother pathways for students. In advising, these measures can guide conversations about which foundational courses create the strongest launchpad for later study, and which advanced courses might benefit from stronger prerequisites to maintain a coherent progression. The promise is insights that translate into healthier curricula and better student outcomes, grounded in a transparent, testable framework.
Finally, the work opens a conversation about modeling curricula in a way that captures their essential character without becoming unwieldy. The authors acknowledge that the KN and ER models offer only rough approximations and that a more faithful generative model remains an open challenge. The real value, for now, is in providing a shared lens and a disciplined vocabulary for comparing curricula at scale. The broader goal is to move from case studies of individual departments to comparative science of education, where we can ask not just what courses exist, but how the architecture of a curriculum shapes the journey of learning itself.
In their own words, the researchers emphasize that course-prerequisite networks crystallize the flow of knowledge across an entire university. By distilling this flow into breadth, depth, and flux, they give educators and policymakers a language to describe the invisible geometry of learning, a geometry that unfolds across dozens of departments, thousands of students, and years of inquiry. The study invites us to imagine not just which courses to offer, but how many layers of understanding a program should build, how far it should push students toward ambitious horizons, and how knowledge should travel through the fabric of the curriculum to reach those horizons. It is a modest, ambitious move toward a science of education that respects both the local decisions of instructors and the global patterns they collectively produce.
