Instructor: Prof. INDIKA Rajapakse (indikar@umich.edu)

Teaching Assistant: RAM Prakash (nuvi@umich.edu) 

Location: 2260 Undergraduate Science Building

Class Time: Tuesday and Thursday, 8:30 AM - 10:00 AM

Office Hours: 

• Indika R - Tuesday and Thursday , 4:00 PM - 5:00 PM:  https://meet.google.com/dnm-zipr-tsb  or in person after class

• Ram P - Wednesday 3:00 - 4:00 PM Zoom and Friday 3:00 PM - 4:00 PM in person Room 3370, NCRC B520

Links

• Syllabus

• Topics and Timeline (Topics and Timeline subject to change without notice. Please check regularly)

• Piazza (Please sign in and add yourself to the course if you have not already) and Course Feedback Form

• Challenge Problem! This is from the book by Smale and Hirsch, Differential Equations, Dynamical Systems, and Linear Algebra. If you solve this problem, you are done for the semester. You can easily publish your answer and become famous! If you would like more challenging problems, please see the following paper. Steve Smale: Mathematical problems for the next century. 

• Simplicity, Rigor, and Magic! and James Simons: My Guiding Principles

• Basic Notation

References of Interest

• Claude Shannon: His famous paper and his thesis

• MATLAB: 1) MATLAB Tutorial 2) Basic Functions Reference

• Great book with codes: Cleve Moler. Numerical computing with MATLAB. Society for Industrial and Applied Mathematics, (2004), Solutions to Exercises

• Digital Library and TwinCell Blueprint: An unofficial digital library I maintain that contains books and papers on topics related to my research and teaching

• Jon Kleinberg's Book: Networks, Crowds, and Markets  and Albert-László Barabási's Book: Network Science

PROBLEM OF THE DAY 

POD 1 (09-01-2026): Introduction

POD2 (09-08-2026): Adjacency Matrix and Graph Laplacian 

POD3 (09-24-2026): PC1 and Fiedler Vector

POD4 (10-08-2026): Definition of Emergence

PROBLEM SETS

This section includes assignments, solutions, and helpful resources. 

• LaTeX Template

• Problem Set 1

  • Preamble Form

  • Constructing a Network from Your Data 

  • Data and Helper Code

  • Solutions and Solution Code

• Problem Set 2

  • Structure and Function of Networks

  • Data

  • Helper Code

  • Solutions

• Problem Set 3

  • Dynamic Mode Decomposition

  • Data Description

  • Helper Code and Data

  • Solutions

• Problem Set 4

  • Part 1: Preamble for the Final Project

FINAL Project: Due December 17, 2025 (no extensions!)

• Teams and Titles

• The Heilmeier Catechism and an Example

• Data

• Latex Template and Some Final Project Examples from Last Year

NOTES, SLIDES, AND PAPERS

Date: 09-01-2026

Quote of the Day

"Optimism is the faith that leads to achievement. Nothing can be done without hope and confidence" ― Helen Keller 

Introduction: Slides

Papers

1. Turing, Alan Mathison. "The chemical basis of morphogenesis." Philosophical Transactions of the Royal Society B: Biological Sciences, 370(1666), p.20140218. (Classic!)

2. Rajapakse I. "Conversation with Dr. Steve Smale and Dr. Lee Hartwell." NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY, 68 (9). 

Date: 09-03-2026

Quote of the Day

"Ideas only realize their power when people understand them" ― Small Worlds: The Dynamics of Networks between Order and Randomness (Book) 

Notes

• Matrix Operations

• Basic Graphs and Adjacency Matrices 

• Networks: (1) Random, (2)  Small-world, (3) Scale-free, (4) Ramanujan 

Papers

1. Small World: Watts, Duncan J., and Steven H. Strogatz. "Collective dynamics of ‘small-world’networks." Nature 393.6684 (1998): 440-442.

2. Scale-free: Barabási, Albert-László, and Réka Albert. "Emergence of scaling in random networks." Science 286, no. 5439 (1999): 509-512. 

3. Small World: Milgram, Stanley. "The small world problem." Psychology today 2, no. 1 (1967): 60-67. 

Date: 09-08-2026

Quote of the Day

"Nature has a great simplicity and therefore a great beauty" ― Richard Feynman 

Notes

• Networks and Linear Algebra

• Singular Value Decomposition (SVD)! This lecture is in memory of Cleve Moler.

Date: 09-10-2026

Quote of the Day

"Greatness comes from character, and character isn't formed out of smart people. It is formed out of people who have suffered" ― Jensen Huang 

Today, we have an in-person visitor from NVIDIA, Dr. Angus Forbes, who will present network visualization methods and NVIDIA tools for digital biology. 

• Data Visualization Resources

• Slides from Angus Forbes

Date: 09-15-2026

Quote of the Day

"The only way to do great work is to love what you do" ― Steve Jobs 

• The Extraordinary SVD again: This lecture is in memory of Cleve Moler. During the lecture, we will watch a recording of Cleve. 

  1. Some Biological Motivations 

  2. Singular Value Decomposition (SVD)

  3. How SVD Saved the Universe: Cleve Moler

Additional Materials from Cleve Moler 

• 4. SVD Measures Partisanship in the U. S. Senate

  5. 1976 Matrix Singular Value Decomposition Film  Direct link to YouTube:  https://www.youtube.com/watch?v=R9UoFyqJca8

Date: 09-17-2026

Quote of the Day

"Science and everyday life cannot and should not be separated" ― Rosalind Franklin 

1. SVD again...

2. The Eckhart-Young Theorem

Papers

• 1. Sweeney P, Chen C, Rajapakse I, Cone R.  "Network Dynamics of Hypothalamic Feeding Neurons." Proceedings of the National Academy of Sciences, 118.14 (2021). 

  2. Lieberman-Aiden, Erez, ..., Groudine Mark, ..., Lander Eric. "Comprehensive mapping of long-range interactions reveals folding principles of the human genome." Science 326.5950 (2009): 289-293.

Date: 09-22-2026

Quote of the Day

"I don't care that they stole my idea . . I care that they don't have any of their own" ― Nikola Tesla

1. Why Low Rank? Biological Motivations 

2. SVD again... (I would really recommend reading pages 7–22 from time to time!) 

Papers

1. Stankovski, Tomislav, Tiago Pereira, Peter VE McClintock, and Aneta Stefanovska. "Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences." Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 377, no. 2160 (2019): 20190039. 

2. Too much Coupling

Date: 09-24-2026

Quote of the Day

"You never lose a dream, it just incubates as a hobby" ― Larry Page

• Network Centrality and Google Matrix

• Laplacian

Papers

1. Brin, Sergey, and Lawrence Page. "The anatomy of a large-scale hypertextual web search engine." (1998). 

PatentsUS6285999B1

1. US6285999B1 Method for node ranking in a linked database. Lawrence Page: 1998-01-09

2. 115-008219-US-PS1 Network approach to navigating the human genome. Indika Rajapakse: 2020.

Date: 09-29-2026

Quote of the Day

"Innovation distinguishes between a leader and a follower" ― Steve Jobs 

Notes

• Network Neuroscience: Some neuroscience networks and relevant papers.

• Poincare Diagram: Stability diagram classifying Poincaré maps as stable or unstable according to their features

• Dynamic Mode Decomposition (DMD), Slides

Book

• Kutz, J. Nathan, et al. Dynamic mode decomposition: data-driven modeling of complex systems. Society for Industrial and Applied Mathematics, 2016. 

Chapter 1: Dynamic Mode Decomposition: An Introduction

Please get familiar with these two papers. They will give you some idea of Network Controllability 

1. Liu, Yang-Yu, Jean-Jacques Slotine, and Albert-László Barabási. "Controllability of complex networks." Nature 473.7346 (2011): 167-173. Slides: Courtesy of Yang Liu

2. Ronquist S, Patterson G, Muir LA, Lindsly S, Chen H, Brown M, Wicha MS, Bloch A, Brockett R, Rajapakse I. "Algorithm for cellular reprogramming." Proceedings of the National Academy of Sciences. 2017 Nov 7;114(45):11832-7. Data-guided Control (DGC). Network Controllability and Applications: Slides from Scott Ronquist 

Date: 10-01-2026

Quote of the Day

"Time isn’t the main thing. It’s the only thing" ― Miles Davis 

Examples to help solidify the concepts of DMD and  Network Controllability

Papers

1. Proctor, Joshua L., Steven L. Brunton, and J. Nathan Kutz. "Dynamic mode decomposition with control." SIAM Journal on Applied Dynamical Systems 15, no. 1 (2016): 142-161. 

2. Liu S, Chen H, Ronquist S, Seaman L, Ceglia N, Chen PY, Higgins G, Baldi P, Smale S, Hero A, Muir LA, Rajapakse I. "Genome Architecture Mediates Transcriptional Control of Human Myogenic Reprogramming." iScience (2018). 

Date: 10-06-2026

Quote of the Day

"Ideas only realize their power when people understand them" ― Duncan Watts's book Six Degrees: The Science of a Connected Age 

• Network Controllability and Data-Guided Control (DGC): I will make this document more concise!

• Network Controllability and Applications: Slides from Scott Ronquist 

Papers: Please check this slide deck. I will update it from time to time when I find a good paper on network control. Thanks. 

1. Liu, Yang-Yu, Jean-Jacques Slotine, and Albert-László Barabási. "Controllability of complex networks." Nature 473.7346 (2011): 167-173. Slides: Courtesy of Yang Liu

2. Ronquist S, Patterson G, Muir LA, Lindsly S, Chen H, Brown M, Wicha MS, Bloch A, Brockett R, Rajapakse I. "Algorithm for cellular reprogramming." Proceedings of the National Academy of Sciences. 2017 Nov 7;114(45):11832-7. Data-guided Control (DGC). Slides from Scott Ronquist 

Date: 10-06-2026

Quote of the Day

"Imagination is the Discovering Faculty, pre-eminently. It is that which penetrates into the unseen worlds around us, the worlds of Science." ―  Ada Lovelace

• Final projects

• My motivations

  • Rajapakse I, and Smale S. "Emergence of Function from Coordinated Cells in a Tissue." Proceedings of the National Academy of Sciences 114.7 (2017): 1462-1467. 

  • Rajapakse I, and Smale S. "The Pitchfork Bifurcation." International Journal of Bifurcation and Chaos 27.09 (2017): 1750132. 

  • Rajapakse I, and Smale S. "Mathematics of the Genome." Foundations of Computational Mathematics 17.5 (2017): 1195-1217. 

• Network Controllability

Date: 10-13-2026

Quote of the Day

"The original question, 'Can machines think?' I believe to be too meaningless to deserve discussion." ―   Alan Turing, Mechanical Intelligence: Collected Works of A.M. Turing 

• TURING System and Bifurcations

• I will discuss pitchfork and Hopf bifurcations in the context of biological networks using the following paper: Rajapakse I, and Smale S. "Mathematics of the Genome." Foundations of Computational Mathematics 17.5 (2017): 1195-1217.

Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations: This is a very nice book, available online through the university. I have attached the first chapter below; please read it when you have time. 

Why Networks?

Papers on Bifurcations in Biological Networks 

1. Timothy Gardner, Charles Cantor, and James Collins. "Construction of a genetic toggle switch in escherichiacoli." Nature, 403(6767): 339, 2000.

2. Michael Elowitz and Stanislas Leibler. "A synthetic oscillatory network of transcriptional regulators." Nature, 403(6767): 335, 2000.

3. Spencer, Sabrina L., et al. "The proliferation-quiescence decision is controlled by a bifurcation in CDK2 activity at mitotic exit." Cell 155.2 (2013)

Date: 10-15-2026

Quote of the Day

"Nothing is more practical than a good theory" ― Ludwing Boltzmann 

In the first part of the lecture I will finish what I started in the previous class. In the second part I will revisit Data-Guided Control (DGC), for which I will introduce some of the underlying optimization techniques. DGC is closely related to Network Control!

• Computing Network Bifurcations from Data: An Example

Date: 10-22-2026

Quote of the Day

"All things flow" ― Heraclitus 

Recap and Missed Sections

Date: 10-27-2026

"Information is the resolution of uncertainty”― Claude Shannon 

• Entropy

• Graph Entropy and Tensor Entropy: Entropy provides a quantitative measure of data’s inherent randomness and information content, informing feature engineering and model optimization!

• Network Robustness: Network robustness is the ability of a network to survive from random failures or deliberate attacks (e.g., removal of nodes or edges). It is intimately related to the underlying network structure/topology.

Papers

1. Chen, Pin-Yu, et al. "Fast incremental von neumann graph entropy computation: Theory, algorithm, and applications." International Conference on Machine Learning. PMLR, 2019. Slides from ICML 2019 

2. Kitano, Hiroaki. "Biological robustness." Nature Reviews Genetics 5.11 (2004): 826-837.

3. MacArthur, Ben D., and Ihor R. Lemischka. "Statistical mechanics of pluripotency." Cell 154.3 (2013): 484-489. (An illustrative example of entropy in biological systems)

Date: 10-29-2025

“I know what I can do, so I never doubt myself.” ― Usain Bolt

• Why Hypergraphs?

• Tensors and Hypergraphs: Notes and Slides on Tensor Basics

  • Other Resources:  Dr. Charles Van Loan (Wonderful introduction to Tensors!)

• HAT: Hypergraph Analysis Toolbox: Multi-correlations and Hyperlink Prediction (Slides courtesy of Can Chen)

• Popular Hypergraphs: STEPHEN WOLFRAM  and Wolfram Physics Project Launch

Papers

1. Bick, Christian, Elizabeth Gross, Heather A. Harrington, and Michael T. Schaub. "What are higher-order networks?." SIAM Review 65, no. 3 (2023): 686-731. 

2. Dotson, Gabrielle A., Can Chen, Stephen Lindsly, Anthony Cicalo, Sam Dilworth, Charles Ryan, Sivakumar Jeyarajan et al. "Deciphering Multi-way Interactions in the Human Genome." Nature Communications 13, no. 1 (2022): 5498. 

3. Surana A, Chen C, Rajapakse I. "Hypergraph Similarity Measures."  IEEE Transactions on Network Science and Engineering,  2022 (Slides)

I know you’re excited about hypergraphs, and here are some more papers to browse!

GENERAL READING

I will add to this list throughout the semester

1. Strogatz SH. "Exploring complex networks." Nature. 2001 Mar;410(6825):268-76.

2. Hartwell, Leland H., et al. "From molecular to modular cell biology." Nature 402.6761 (1999): C47-C52.

3. Network Thinking: Melanie Mitchell

4. Making connections