Network Analysis of the Paris and Tokyo Subway Systems

Advisor: Dr. Alexis Byers

This thesis applies network analysis to the subway systems of Paris and Tokyo, with the goal of improving subway efficiency and sustainability through insights gained from network science. The paper covers the mathematical preliminaries of graph theory and various topics within network science, focusing particularly on centrality measures. Using networks based on the Paris and Tokyo subway systems, the paper finds that the chromatic number of both networks is 3. Comparisons are made using various centrality scores and descriptive metrics,  the practical implications of the findings are discussed. The results suggest that the insights gained from network analysis can be used to enhance the efficiency and sustainability of subway systems, offering practical applications for public transportation management.

Relevant Links:

• Publication

• Defense (Recording)

• Defense (Slides)

Spring of 2023

We collaborated with Central Mutual Insurance Companies to perform social network analysis as part of The Data Mine project at Purdue University. The team consisted of around 20 students with varying levels of skill. We divided into sub-teams and I had the opportunity to work on the Instagram team. Together with four others, we created a Python program that manually web-scrapes Instagram to develop a network based on an account's followers. Our program can currently go two levels deep (followers of followers) and can be easily adapted to go deeper. Through our analysis, we provided Central with feedback on the most influential accounts and communities in their Instagram network, which they found useful. We traveled to Purdue University to present our work at the The Data Mine Corporate Partners Symposium in April of 2023.

Relevant Links:

• Code (Github)

• Poster and video 

Spring of 2022

The goal of this project was to determine the occurrence of sleep apnea by analyzing biometric data, including heart rate, chest volume, and blood oxygen concentration, recorded from a single individual over an 8-hour period. The individual had a known sleep apnea condition, and the study required further analysis to accurately extrapolate sleep apnea incidents. Models were created for each variable separately using a small window of the data. The heart rate followed an AR(5) model, the chest volume an ARIMA(2,0,10)×(0,1,2) model, and the blood oxygen concentration an ARMA(5,8) model. However, these models revealed non-normally distributed residuals, with the exception of blood oxygen concentration, which also exhibited dependently distributed residuals. The use of stationary models failed to replicate the variation observed in the biometrics. Spectral analysis was recommended to analyze heart rate and blood oxygen concentration, while a different model was needed to examine chest volume. 

Relevant Links:

• Research Paper

Spring of 2022

Our task involved classification, where we used metrics from several Polish companies to predict whether they were likely to go bankrupt. We obtained the data from a global market database, which provided information on bankrupt companies dating back to 2000 (until 2012), and on companies still in operation dating back to 2007 (until 2013). To evaluate our models, we used the F1 score, as specified by the competition guidelines. The model with the highest F1 score was chosen. We attempted to fit multiple models using R. We used decision trees, best subsets selection, forward step-wise selection, and backward step-wise selection methods.

Relevant Links:

• Research Paper

Fall of 2021

In this study we used data collected from the National Survey on Drug Use and Health (2019)  to examine the effect of alcohol usage on the academic performance of students. We used IBM SPSS to do all of our analysis. We used the student's grade (A, B, C, D, and F) as our dependent variable and if someone has every drank alchol as our explanatory variable. We restricted our age group to the category 12 - 17 years old. First, we used cross-tabulation to find that 54% of males and females said they have drank alcohol, this was comprised of 23% for male and 31% for females. We then a Pearson Chi-Square test to determine the risk estimation in two categories. We found that the odds of a males drinking alcohol is around 0.739 times the odds of females and that the odds of a student who obtained a grade of A to drink alcohol is around 0.781 times the odds of those who obtained other grades. Finally we used multinomial logistic regression analysis to compare the adjusted odds ratios of between the grade groups. We found that students who have drunk alcohol are less likely than students who have not to be in the A grade group than the “D or Less than D” grade group with an adjusted odds ratio of 0.301. Students who have drunk alcohol are less likely than students who have not, to be in the B grade group than the “D or Less than D” grade group with an adjusted odds ratio of 0.436. Finally, students who have drunk alcohol are less likely than students who have not, to be in the C grade group than the “D or Less than D” grade group with an adjusted odds ratio of 0.504. 

Relevant links:

• The Effect of Alcohol Usage on Students' Academic Performance in the US (Paper) 

Fall of 2021

In this study I used data collected from the National Survey on Drug Use and Health (2019) and IBM SPSS to examine the likelihood of marijuana usage based on an individual’s gender, age, mental health access, poverty level, and education level. To analyze this data, first basic descriptive analysis was used. Based of the results from this analysis, Chi-Square tests were used and odds ratios were considered. It has been found that males are 1.251 times more likely to use marijuana than females. Also, individuals who have received mental health care are 2.126 times more likely to use marijuana than individuals who have not received mental health care. 

Relevant links:

• Evaluation of Possible Influences on Marijuana Use (Paper) 

Fall 2022

The paper "Exploration of Spectral Graph Theory" delves into the ideas of spectral graph theory necessary to analyze networks based on the Paris and Tokyo subway systems. The study begins with a historical overview of transportation problems in graph theory and proceeds to explore the relationship between linear algebra and graph theory. The study then investigates the preliminaries of linear algebra and graph theory required to analyze the networks. Prior to utilizing Gephi and Python to analyze the networks, the study examines a non-trivial graph by performing manual calculations. This project provides an opportunity to apply mathematical concepts to a practical problem and to appreciate the significance of bridging the gap between mathematical theory and its applications. 

Relevant Links

• Exploration of Spectral Graph Theory (Paper)

• github.com/travisschauer/EigenvectorCentrality 

Spring of 2021 - Winter of 2021

The paper "A Look into Graph Colorings" examines various aspects of graph colorings. The study presents original work on total colorings and entire colorings. Additionally, the paper discusses applications of vertex colorings, including a Python program that applies vertex colorings to solve Sudoku puzzles. The study lays the groundwork for future research in the field of graph colorings. 

Relevant Links:

• A Look into Graph Colorings (Paper) 

• github.com/travisschauer/SodukuSolver 

Spring of 2020

The project involved the application of mathematical, programming, data science, and GIS skills to optimize snowplow routes in Warren, Ohio. A significant amount of research was conducted on route optimization, with a focus on graph theory, particularly Euler Paths and the Chinese Postman Problem. Using Excel and ArcGIS, the primary and secondary snowplow routes were optimized. We then employed graph theory concepts to optimize the routes for the remaining sections of the city. As a result, we were able to update the old routes and create a new checklist that enables the city to run its snowplows more efficiently. 

Relevant links:

• Warren Snowplow (Paper and Presentation) 

• Analyzed Data 

Spring of 2021

In this paper I explored the history of cryptography and reviewed the mathematical background necessary to study the RSA Cryptosystem. I also presented a program written using Visual Basic that demonstrates the RSA Cryptosystem. This paper served as my Senior Capstone and remains one of my favorite projects.

Relevant links:

• A Look into Cryptography (Paper and Presentation) 

• https://github.com/travisschauer/RSAMethods 

Fall of 2019

In this project I used skills in PHP, HTML, and SQL to create an online bicycle store. This website allows users to creates an account, delete their account, track, and edit orders. Customers are able spec out a customer build bicycle and select a delivery option. Site admins are able to edit the status of any order and available parts for selection. 

Relevant Links:

• https://github.com/travisschauer/OnlineBicycleShop 

Spring of 2019

In this project we used Python and XPP-Aut to analyze differential equations and their bifurcations. We investigated the Morris-Lecar model of neuron excitability. We then applied this data to determine various parameters that create certain types of neuron bursting.