We are surrounded by millions of data points each day: the number of steps we walk, changes in temperature, the fluctuations of the stock market. When we're inundated with so much information, how can we determine which of those points, and the networks they belong to, are the most important?
That's what Linda ’23 wanted to know when she began writing an analytical paper at a mentorship program for young STEM researchers. She's since won a MathWorks First Place Prize at the Massachusetts Science & Engineering Fair for her research, which uses analytic tools from algebraic topology to examine the ever-changing global economy.
Connecting the Dots
Linda's research topic might seem purely theoretical, but, she explains, we encounter its underlying logic whenever we engage with a network of connections. Case in point: social media. "On Instagram, each person is friends with another person. And if two people are friends with each other, we can think of the people as nodes, and if they're connected, then you can draw an edge"—a visual marker—"between them." In the same way that Instagram users are linked, points of statistical data can also be connected to each other.
Data doesn't drift in a vacuum, of course. We might wonder which nodes play the most important role in a particular network—for example, to build on Linda's social media analogy, which accounts have the greatest influence on Instagram. One way to do that is to find the centrality measure, which assigns a numerical ranking to each node in a network based on its importance.
But, Linda points out, it can be hard to figure out which methods of ranking we should use to select nodes. Should we pick which social media users to analyze in our example based on follower count or some other quantitative measure? In any complex system, there are hundreds of points to consider, and that's the kind of problem that drew Linda in.
The Research Experience
Linda was introduced to topological data analysis, or TDA, at last year's MIT Program for Research in Mathematics, Engineering, and Science for High School Students (PRIMES), an after-school program matching students with MIT researchers to examine problems in STEM fields. Linda's mentor pitched the idea of a TDA project, and after the pair went through the topic together through background information and lectures, she gave Linda a chance to explore data for herself.
"My research was on developing a new network measure specifically for directed networks"—that is, networks that can be visually represented with arrows between points—"and networks that are really highly clustered and complex," Linda says. "In that case, a lot of the previous existing centrality measures wouldn't work anymore."
Related: Linda Reflects on Commonwealth's "Math Community"
The mathematical problem-solving of developing a new centrality measure made up the first part of Linda's research. The second was to test the measure on a real-world network to prove its efficacy, and Linda opted for the global economy: a system with hundreds of thousands of interconnected nodes to analyze. "Each country is linked with almost every other country, and it is also directed because you have imports and exports," Linda observes. To narrow the window of focus, "I looked at the Asian trade network during the financial crisis."
After learning about the Massachusetts Science & Engineering Fair from the MIT PRIMES directors, who encourage students to exhibit their research wherever possible, Linda was nominated by Commonwealth as the school's representative. She presented before two sets of judges online and submitted a thirty-page paper with the measure she'd developed and its economic application.
"It was an enjoyable experience presenting my research, especially when the judges shoot you questions about your research after your presentation," she says. And Linda credits her time at Commonwealth, where she had the chance to take a topology class as a junior, for nurturing her math enthusiasm. She's known around school as a dedicated member of the Math Team, and earned a spot at the American Mathematics Competition earlier this year.
"I am really grateful for the school offering me this opportunity to participate in the competition, and also offering a lot of advanced and cool math and science courses for me and other students," Linda says. "I think without these, I wouldn't be able to follow my passion for research and math and science as smoothly."