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Course Description

This course will explore methods of applied mathematics used in network science. We will alternate between three class modes to (1) cover fundamental mathematical background that sets us up to (2) read how those fundamentals have been used in the networks context and (3) apply network calculations on data from different application areas. The first mode will draw largely from material in standard texts, the second will look at specific research articles from the applied math literature, and the third will work through computational primers so you leave the course being able to run some network analyses yourself.

Prerequisites

There are no official prerequisites, but students need to be able and willing to engage with the material at a graduate level. The course material relies heavily on various components of Linear Algebra, Scientific Computation, Probability, and other traditional Methods of Applied Mathematics, and students must be willing to expand their knowledge in these and other areas of mathematics as needed. Advanced undergraduates may be allowed with permission of the instructor.

Learning Outcomes

Students will become familiar with a variety of methods and measures of network science and develop experience performing their own network analyses of data.

Expectations and Grading

Students will be asked to actively participate in classroom discussions about the articles assigned as readings ahead of class. There will also be occasional short computational assignments. Together these will comprise 1/3 of the course grade. By the end of the term, students will carry out a computational project applying methods to a selected data set (1/3 course grade) and write a report about another paper selected from the literature (1/3 course grade).

Computational Notebooks

Written computational assignments must be completed in computational notebooks combining descriptive narrative text with embedded code and output (figures and tables). You are welcome to use any computational environment you prefer, as long as it is performed in the form of a notebook (e.g., MATLAB Live Editor, R Notebook, Jupyter notebook for Python or Julia). Paying attention to effective communication is an essential element to properly convey the method and value of a data analysis. As such, grades will be as much about the quality of the communication as that of the computation.

Academic Honesty

The Academic Honor Principle is an essential tenet of the Dartmouth community. Collaboration is strongly encouraged in this course. Ultimately, all assignments and reports submitted must represent your own understanding of the material.

Schedule and X Hours

Our class meeting time will be shortened on 9/23 and cancelled on 11/16. To make up this time we will hold regular class meetings on 10/1, 10/22, and 11/12.

Attendance

You are expected to attend class in person unless you have made alternative arrangements due to, e.g., illness, medical reasons, or the need to isolate due to COVID-19. For the health and safety of our class community, please: do not attend class when you are sick, nor when you have been instructed by Student Health Services to stay home. Please contact me to discuss how to best catch up on any course material that you miss.

Religious Observances

Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please notify me well in advance to discuss appropriate accommodations.

Safety

In accordance with current College policy, all members of the Dartmouth community are required to wear a suitable face covering when indoors, regardless of vaccination status. This includes our classroom and other course-related locations, such as labs, studios, and office hours. If you need to take a quick drink during class, please dip your mask briefly for each sip. Eating is never permitted in the classroom. (The only exception to the mask requirement is for students with an approved disability-related accommodation; see below.) If you do not have an accommodation and refuse to comply with masking or other safety protocols, I am obligated to assure that the Covid health and safety standards are followed, and you will be asked to leave the classroom. You remain subject to course attendance policies, and dismissal from class will result in an unexcused absence. If you refuse to comply with masking or other safety protocols, and to ensure the health and safety of our community, I am obligated to report you to the Dean’s office for disciplinary action under the Guarini School’s Standards of Conduct. Additional COVID-19 protocols may emerge. Pay attention to emails from the senior administrators at the College.

Student Accessibility and Accommodations

Students requesting disability-related accommodations and services for this course are required to register with Student Accessibility Services (SAS; Getting Started with SAS webpage; student.accessibility.services@dartmouth.edu; 1-603-646-9900) and to request that an accommodation email be sent to me in advance of the need for an accommodation. Then, students should schedule a follow-up meeting with me to determine relevant details such as what role SAS or its Testing Center may play in accommodation implementation. This process works best for everyone when completed as early in the quarter as possible. If students have questions about whether they are eligible for accommodations or have concerns about the implementation of their accommodations, they should contact the SAS office. All inquiries and discussions will remain confidential.

Diversity and Inclusion

I strongly agree with the sentiments expressed in this sample syllabus statement on diversity from Monica Linden, Senior Lecturer in Neuroscience at Brown University. In linking to this statement, rather than copying it, I aim to both acknowledge the source and, in asking you to follow the link, highlight the message therein. In particular, I would like to emphasize that we will make an effort to read papers from a diverse group of scientists, while acknowledging that limits still exist on this diversity. I encourage you to talk to me if anything in class or out of class makes you uncomfortable or if you have any suggestions to improve our environment or the quality of the course materials.

Mental Health and Wellness

The academic environment at Dartmouth is challenging, our terms are intensive, and classes are not the only demanding part of your life. There are a number of resources available to you on campus to support your wellness, including the undergraduate deans, Counseling and Human Development, and the Student Wellness Center. I encourage you to use these resources to take care of yourself throughout the term, and to come speak to me if you experience any difficulties.

Title IX

At Dartmouth, we value integrity, responsibility, and respect for the rights and interests of others, all central to our Principles of Community. We are dedicated to establishing and maintaining a safe and inclusive campus where all have equal access to the educational and employment opportunities Dartmouth offers. We strive to promote an environment of sexual respect, safety, and well-being. In its policies and standards, Dartmouth demonstrates unequivocally that sexual assault, gender-based harassment, domestic violence, dating violence, and stalking are not tolerated in our community.

The Sexual Respect Website at Dartmouth provides a wealth of information on your rights with regard to sexual respect and resources that are available to all in our community.

Please note that, as a faculty member, I am obligated to share disclosures regarding conduct under Title IX with Dartmouth's Title IX Coordinator. Confidential resources are also available, and include licensed medical or counseling professionals (e.g., a licensed psychologist), staff members of organizations recognized as rape crisis centers under state law (such as WISE), and ordained clergy (see https://dartgo.org/titleix_resources).

Should you have any questions, please feel free to contact Dartmouth's Title IX Coordinator or the Deputy Title IX Coordinator for the Guarini School. Their contact information can be found on the sexual respect website at: https://sexual-respect.dartmouth.edu.

Schedule

schedule.txt

References

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Bertozzi, A., and A. Flenner. “Diffuse Interface Models on Graphs for Classification of High Dimensional Data.” SIAM Review 58, no. 2 (January 1, 2016): 293–328. https://doi.org/10.1137/16M1070426.

Bryan, Kurt, and Tanya Leise. “The $25,000,000,000 Eigenvector: The Linear Algebra behind Google.” SIAM Review 48, no. 3 (January 1, 2006): 569–81. https://doi.org/10.1137/050623280.

D’Souza, Raissa M., and Jan Nagler. “Explosive Percolation: Novel Critical and Supercritical Phenomena.” Nature Physics 11, no. 7 (July 1, 2015): 531–38. https://doi.org/10.1038/nphys3378.

Durrett, Rick. Random Graph Dynamics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press, 2007. https://doi.org/10.1017/CBO9780511546594. https://services.math.duke.edu/~rtd/RGD/RGD.pdf.

Durrett, Richard, James P. Gleeson, Alun L. Lloyd, Peter J. Mucha, Feng Shi, David Sivakoff, Joshua E. S. Socolar, and Chris Varghese. “Graph Fission in an Evolving Voter Model.” Proceedings of the National Academy of Sciences 109, no. 10 (March 6, 2012): 3682–87. https://doi.org/10.1073/pnas.1200709109.

Estrada, Ernesto, and Desmond J. Higham. “Network Properties Revealed through Matrix Functions.” SIAM Review 52, no. 4 (2010): 696. https://doi.org/10.1137/090761070.

Fosdick, Bailey K., Daniel B. Larremore, Joel Nishimura, and Johan Ugander. “Configuring Random Graph Models with Fixed Degree Sequences.” SIAM Review 60, no. 2 (January 1, 2018): 315–55. https://doi.org/10.1137/16M1087175.

Gleich, David F. “PageRank Beyond the Web.” SIAM Review 57, no. 3 (January 1, 2015): 321–63. https://doi.org/10.1137/140976649.

Grindrod, P., and D. Higham. “A Matrix Iteration for Dynamic Network Summaries.” SIAM Review 55, no. 1 (January 1, 2013): 118–28. https://doi.org/10.1137/110855715.

Grindrod, Peter, Mark C. Parsons, Desmond J. Higham, and Ernesto Estrada. “Communicability across Evolving Networks.” Physical Review E 83, no. 4 (April 25, 2011): 046120. https://doi.org/10.1103/PhysRevE.83.046120.

Higham, Desmond J. “Centrality-Friendship Paradoxes: When Our Friends Are More Important than Us.” Journal of Complex Networks 7, no. 4 (August 1, 2019): 515–28. https://doi.org/10.1093/comnet/cny029.

Hinch, E. J. Perturbation Methods. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press, 1991. https://doi.org/10.1017/CBO9781139172189.

Holme, Petter, Mason A. Porter, and Hiroki Sayama. “Who Is the Most Important Character in Frozen? What Networks Can Tell Us About the World.” Frontiers for Young Minds. Accessed September 15, 2021. https://kids.frontiersin.org/articles/10.3389/frym.2019.00099.

Kolaczyk, Eric D. Statistical Analysis of Network Data: Methods and Models, 2009.
http://dx.doi.org/10.1007/978-0-387-88146-1.
Available online through Library:
https://search.library.dartmouth.edu/permalink/01DCL_INST/16rgcn8/alma991033590274005706

Kolaczyk, Eric D. and Csardi, Gabor. Statistical Analysis of Network Data with R. Second Edition. 2020.
https://doi.org/10.1007/978-3-030-44129-6.
Available online through Library:
https://search.library.dartmouth.edu/permalink/01DCL_INST/16rgcn8/alma991033590872505706
Code: https://github.com/kolaczyk/sand.

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Porter, Mason, and James Gleeson. Dynamical Systems on Networks: A Tutorial. Vol. 4. Frontiers in Applied Dynamical Systems: Reviews and Tutorials. Cham: Springer International Publishing, 2016. https://doi.org/10.1007/978-3-319-26641-1.  Available online at https://www.math.ucla.edu/~mason/papers/frontiers-published-tutorial.pdf.

Priebe, Carey E., Youngser Park, Joshua T. Vogelstein, John M. Conroy, Vince Lyzinski, Minh Tang, Avanti Athreya, Joshua Cape, and Eric Bridgeford. “On a Two-Truths Phenomenon in Spectral Graph Clustering.” Proceedings of the National Academy of Sciences 116, no. 13 (March 26, 2019): 5995–6000. https://doi.org/10.1073/pnas.1814462116.

Ramani, Arjun S., Nicole Eikmeier, and David F. Gleich. “Coin-Flipping, Ball-Dropping, and Grass-Hopping for Generating Random Graphs from Matrices of Edge Probabilities.” SIAM Review 61, no. 3 (January 1, 2019): 549–95. https://doi.org/10.1137/17M1127132.

Taylor, Dane, Sean A. Myers, Aaron Clauset, Mason A. Porter, and Peter J. Mucha. “Eigenvector-Based Centrality Measures for Temporal Networks.” Multiscale Modeling & Simulation 15, no. 1 (January 1, 2017): 537–74. https://doi.org/10.1137/16M1066142.

Watts, Duncan J., and Steven H. Strogatz. “Collective Dynamics of ‘Small-World’ Networks.” Nature 393, no. 6684 (June 1998): 440–42. https://doi.org/10.1038/30918.