by Adrienne Berard

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April 28, 2021

Singer, songwriter, producer, visual artist and two-time Grammy winner Frank Ocean can now say math proves he is the most influential artist of the 21^st century.

Ocean can give their new status to three undergraduates from William & Mary, whose mathematical model analyzing musical influence recently earned the trio the title of “Outstanding Winner” in the International Mathematical Modeling Competition (MCM) led by COMAP.

COMAP is the Consortium for Mathematics and its Application, a non-profit organization dedicated to the advancement of mathematics education for all ages. Each year, COMAP sponsors a competition, offering a set of open mathematical problems to analyze.

This year’s competition took place over virtually a single weekend, February 4-8, during which teams of three students researched, modeled, and wrote a solution to an open, interdisciplinary modeling problem.

A total of 16,059 teams participated in MCM’s interdisciplinary modeling competition and only 19 were selected as “outstanding winners”. William & Mary was the only team outside of China to receive such a designation.

William & Mary was also named one of the top four teams in the entire competition, earning them the International COMAP Scholarship Award, a $ 9,000 scholarship to be divided among team members. and $ 1000 to the school represented. William & Mary was also named one of the top teams by the Mathematical Association of America (MAA) and the Society for Industrial and Applied Mathematics (SIAM).

The three members of the William & Mary team, Clare Heinbaugh ’23, Stuart Thomas ’21 and Ethan Shelburne ’21, are all 1693 Fellows, an elite research-oriented group of four years selected from among the best and the brightest of the William & Mary pool of candidates.

Shelburne was a member of last year’s team, which also won the title of “Outstanding Winner” and received the Leonhard Euler Award and the $ 10,000 COMAP scholarship.

“These are back-to-back wins for us,” said Anh Ninh, assistant professor in the mathematics department at W&M who advised this year’s team. “I hope we can carry on that tradition and that more students on campus – not just math – will enter data analysis and modeling competitions like this one because they are so interdisciplinary in nature. Having people from diverse fields like math, IT, and business will create great teams for us in the future. “

Thomas and Shelburne first entered the competition as freshmen, earning the first year “successful participant” and the “honorable mention” when they entered second year. Now the William & Mary team of which Shelburne has been a part for the four years has twice won “an outstanding winner”.

“I am delighted to have won this competition for a second year, and I am very grateful to my brilliant teammates; loving both music and math really paid off this time around, ”said Shelburne, who describes himself as a“ music nerd ”and will be attending graduate school in math at the University of British Columbia.

Heinbaugh said that despite his lack of experience with competition, Thomas and Shelburne made him feel like an integral part of the team.

“I wanted to do my best and build on my strengths: coding and organizing,” said Heinbaugh, who specializes in computer science and used Python and neural networks to aggregate data and validate metrics for it. ‘team.

Shelburne and Thomas say they have seen themselves grow up as scholars and writers over the past four years at William & Mary.

“I am honored that this result reflects this progress,” said Thomas, a physics student who will be undertaking a doctorate. program at UMD College Park next year, focusing on condensed matter theory. He is also a member of Syndicate, a campus hip-hop dance group, as well as a member of the W&M Club fencing team and the Symphony Orchestra, where he plays the trombone.

“I really enjoyed combining my love of music with a love of math for this year’s competition,” he said. “It’s a relationship that I don’t often get to explore.”

Problem D

According to the competition’s website, the interdisciplinary modeling competition “is designed to develop interdisciplinary problem-solving skills in science, technology, engineering, mathematics (STEM) and the humanities, as well as data science and humanities skills. written communication.”

Candidates choose a problem from a set of problems that require data analysis, creative modeling, and scientific methodology to solve, as well as effective writing and visualization to communicate.

The William & Mary team selected Problem D, the Operations Research / Network Science Problem, which is generally the type of problem-solving taught in W & M’s Computational Operations Research program. This year, Problem D was to analyze the musical influence of artists and how it can show up in the attributes of the music they create.

The team received four datasets from the Integrative Collective Music Society. The students analyzed data on the music of 5,854 artists to develop a model to measure the influence of artists and track the similarity of music. They used these two measures to understand musical evolution, to analyze the musical influence between artists and to identify artistic revolutionaries.

The judges noted in a press release that due to the multidisciplinary nature of the issues, the teams solved these issues using a wide variety of methods and tools.

The William & Mary team built a data-driven model capable of analyzing musical influence in terms of genre and content over the past 90 years, dividing each decade into its own musical period. The team devised a quantitative way to assess both the direct and indirect influence of any artist, as well as the effect an artist has on their own genre and others. They used a combination of four factors, all based on a network of influences reported by the artists.

“Three of them are variations of ‘Katz centrality’, a concept in graph theory that quantifies how central a given artist is to the network,” Thomas explained. “Direct network centrality measures immediate influences, indirect network centrality measures deep influences, direct gender centrality measures influence on an artist’s genre, and gender entropy measures the variability of influence. of an artist. As an extension of a previous study, we use the concept of entropy in physics to measure the “chaos” of an artist’s influence.

Using all of these determinants, the team found Frank Ocean to be the most influential artist of the 21st century. The team viewed both direct and indirect influence as a way to measure the impact of a single artist on the music network as a whole. In their winning article, the team sets the example that many artists consider Elvis Presley to be a major influence.

“[They] no doubt remember his hit song “Hound Dog”. However, “Hound Dog” was written by lesser-known artist Big Mama Thornton. So any follower of Elvis was probably also influenced by Big Mama Thornton, even if they don’t cite her as an influence, ”they write.

Sonic and iconic

In trying to break down the data further into gender, the team encountered a problem. They found that gender did not predict or correlate with musical elements, so in an original solution, the team created their own categories based solely on auditory data.

The team developed four “Sonic Genres,” named after famous songs that pretty much represent the sound of each category as a whole. They created the genre “Here Comes the Sun” for folk music, the genre “Thunderstruck” for high energy music, the genre “I Will Survive” for dance music and the genre “Swan Lake” for instrumental music. . They then trained a sequentially modeled neural network to predict gender from sound variables, rather than established genres.

“We could thus create detailed and parallel stories of musical evolution in terms of both the genres we know and the ‘sound genres’ that emerge naturally from sound content alone,” they write. “With these tools, we believe we can identify and uncover many fascinating periods of change; maybe one day Barry White will be seen as as revolutionary as the Beatles.

After developing tools to analyze musical influence and sound similarity across the entire network and subnets of their model, the team then began to classify musical revolutions.

The team’s model suggests that the 1930s were the period of most significant change in the sonic properties of music in the past 90 years. The team identified blues artist Howlin ‘Wolf and jazz artist Nat King Cole as two of the greatest revolutionaries who began during this time.

“We did our best to back up our choices with substantive research, but the final list of artists considered influential and groundbreaking was impossible to validate against a ground truth,” the team wrote. “This speaks to the nature of the use of mathematics to determine musical revolutionaries and their influence; some elements will always be subjective and difficult to quantify fairly. “