Urban Street Network Topology and City Scaling

ISEF Category: Mathematics

Subcategory: Geometry and Topology  ·  Difficulty: Advanced  ·  Setup: Home Setup  ·  Time: Full Year

The Hook

City maps are not just pictures, they are math objects. Every intersection, block, and loop changes the shape of the network. You can measure those patterns with topology, the math of connectedness and holes. That lets you compare cities in a way that goes beyond area and population.

What Is It?

This project asks a simple question with a deep math answer, what does a city look like when you treat its streets as a network? You turn roads into edges, intersections into nodes, and closed loops into holes. Then you count features like the Euler characteristic and the first Betti number. The Euler characteristic is a summary number for shape, and the first Betti number counts independent loops. Think of it like checking how many ways a network can circle back on itself without retracing the same path.

You can compare many cities and see whether these numbers follow scaling laws. A scaling law looks for patterns like, as a city gets larger, does its loop structure grow in a predictable way? If your data fit a power law, you can test whether the relationship is strong or just noise. This turns a map project into a real study of urban form.

Why This Is a Good Topic

This is a strong science fair topic because you can get real data, build a clear math model, and test a hypothesis across many cities. OpenStreetMap gives you free street data, so you do not need a lab. The project connects topology to urban planning, traffic flow, and city design, which makes the results easy to explain. You can learn data cleaning, graph theory, curve fitting, and significance testing in one project.

Research Questions

• How does the first Betti number scale with city population across a set of 100 or more cities?
• How does the Euler characteristic change with road density in cities of different sizes?
• What is the effect of neighborhood boundary size on the measured loop count in a city street network?
• Does a power-law model fit the relationship between city area and first Betti number better than a linear model?
• To what extent do grid-like cities differ from organic street networks in their topology metrics?
• Which topology measure, Euler characteristic, first Betti number, or cycle density, best separates cities by region or growth pattern?

Basic Materials

• Laptop or desktop computer with enough storage for map files.
• Internet access for downloading OpenStreetMap data.
• Spreadsheet software or Python with basic data analysis libraries.
• OpenStreetMap exports or city boundary files.
• A list of at least 100 cities with consistent selection rules.
• External drive or cloud storage for backups.
• Notebook for tracking data cleaning decisions.

Advanced Materials

• High-performance laptop or access to a university workstation.
• Python with NetworkX, pandas, NumPy, SciPy, and Matplotlib.
• QGIS or another GIS tool for map clipping and boundary checks.
• OpenStreetMap street network extracts or regional graph datasets.
• City boundary shapefiles or GeoJSON files.
• Version control system such as Git for tracking code changes.
• Statistical testing tools for regression diagnostics and model comparison.

Software & Tools

• Python: Builds street-network graphs, computes topology metrics, and fits scaling models.
• NetworkX: Turns map data into graphs and helps count nodes, edges, and cycles.
• pandas: Organizes city-level measurements and cleans your dataset.
• QGIS: Checks city boundaries and helps clip map data to the same study area.
• GeoPandas: Links geographic shapes to your numeric network data.

Experiment Steps

1. Define the city set and write rules for which street networks count as comparable.
2. Choose one map representation, then keep node, edge, and boundary rules consistent across every city.
3. Decide how you will compute Euler characteristic, first Betti number, and any normalized network measures.
4. Build a clean dataset, then document every city you exclude and why.
5. Fit your scaling model, then compare power-law, linear, and log-linear options.
6. Test whether your pattern stays strong after you change boundary rules or city-selection rules.

Common Pitfalls

• Mixing city boundary definitions, which changes the network size and makes comparisons unfair.
• Counting disconnected road fragments as part of the main network, which inflates loop measures.
• Using different map extraction rules for different cities, which creates fake trends.
• Fitting a power law without checking alternatives, which can make an ordinary curve look special.
• Ignoring outlier cities with unusual road grids or coastlines, which can distort the scaling exponent.

What Makes This Competitive

A strong version of this project does more than plot a trend. You test several models, report confidence intervals, and show that your result survives different boundary choices. You can also compare city types, like planned grids versus older organic street layouts. That kind of analysis shows real control over the math, not just data collection.

Project Variations

• Compare downtown cores instead of whole cities to see whether topology changes with scale.
• Analyze pedestrian paths, bike networks, or transit-connected street graphs instead of car roads.
• Test whether coastal cities, inland cities, or river cities show different loop-scaling patterns.

Learn More

• OpenStreetMap Wiki: Learn how map data are structured and where to find city extracts.
• NetworkX Documentation: Read how to build and analyze graphs in Python.
• MIT OpenCourseWare, Introduction to Graph Theory: Review graph basics, including paths, cycles, and connectivity.
• US Census Bureau Geography Resources: Find boundary data and learn how city limits are defined.
• Elements of Discrete Mathematics: Use a library copy or preview to review graph theory and counting methods.