1.022 | Fall 2018 | Undergraduate

# Introduction to Network Models

## Calendar

LEC # TOPICS DUE DATES
1

Course speci fics, motivation, and intro to graph theory

• Course specifi cs: times, office hours, homework, exams, bibliography, etc.
• General motivation: What are networks? What is network science? Impacts, ubiquity, historical background, examples.
• Course description and contents: A quick overview of the things that we are going to learn.
• Basic graph theory: vertices, edges, directed graphs, simple graphs, weighted graphs, neighborhoods, degree, path, cycle.

2

Introduction to graph theory

• More on graph theory: Connectivity, components, giant components, distance, small-world phenomenon, adjacency and incidence matrices.

3

Strong and weak ties, triadic closure, and homophily

Homework 1 distributed

4

Centrality measures

• Detection and identi cation of important agents.
• Degree, closeness, betweenness, eigenvector, and Katz centrality.

5

Centrality and web search, spectral graph theory

• Page rank and web search.
• Eigenvalues and eigenvectors of graph matrices and their properties.
• Quadratic forms on graphs and Laplacian.

6–7

Spectral graph theory, spectral clustering, and community detection

• Properties of graph Laplacian.
• Derive spectral clustering formulation as a relaxation of modularity maximization.
• Community detection using ratio cut criterion.

Homework 2 distributed during Lecture 6

Homework 1 due by Lecture 6
8–10

Network models

• Graphs as realizations of stochastic processes: Introduce the general idea.
• Erdős-Rényi graphs, branching processes. De nition, examples, phase transition, connectivity, diameter, and giant component.

Homework 3 distributed during Lecture 10

Homework 2 due by Lecture 9
11

Con figuration model and small-world graphs

• Con guration model, emergence of the giant component.
• Small-world graphs: De nition from rewiring a regular graph, balance between clustering coefficient and network diameter.

12

Growing networks

• Growing networks.
• Preferential attachment and power laws: the rich get richer effect. Degree distribution observed in real life, example of a dynamic generative process leading to this distribution, mean fi eld analysis.

Homework 3 due

Midterm exam

Homework 4 distributed

13–14

Linear dynamical systems

• Convergence to equilibrium.
• Stability, eigenvalue decomposition, Lyapunov functions.

Homework 5 distributed during Lecture 14

Homework 4 due by Lecture 14
15

Markov chains

• Perron-Frobenius theorem.
• Random walk on graphs.

16–17

• Distributed computation.
• Markov chain convergence and Cheeger’s inequality.

Homework 6 distributed during Lecture 17

Homework 5 due by Lecture 16
18–19

Learning and herding

• Simple Herding Experience.
• Aggregate Beliefs and the “Wisdom of Crowds.”
• The DeGroot Model: The seminal network interaction model of information transmission, opinion formation, and consensus formation.

20

Epidemics

• Models of diffusion without network structure: Bass model
• Models of diffusion with network structure
• The SIR Epidemic Model
• The SIS Epidemic Model

Homework 6 due
21

Introduction to game theory I

• Game theory motivation: Decision-making with many agents, utility maximization.
• Basic ingredients of a game: Strategic or normal form games.
• Strategies: Finite / Infi nite strategy spaces.
• Best responses, dominant, and dominated strategies.
• Iterated elimination of dominated strategies and dominant solvable games.
• Nash equilibrium.

Homework 7 distributed

Part I descriptive write-up of final project (non-default version) due
22

Introduction to game theory II

• Nash equilibrium: more examples.
• Multiplicity of equilibria.
• Pareto-Optimality and social optimality: Price of anarchy.
• Nonexistence of pure strategy Nash equilibria with a touch of mixed strategies.
• Fixed point theorems and existence of Nash equilibrium in infi nite games.

Get familiar with the data and hand in your results for Part I of the final project (default version)
23

Application of game theory to networks

• Traffic equilibrium: non-atomic traffic models.
• Socially-Optimal routing and inefficiency of equilibrium.

Homework 8 distributed

Homework 7 due
24 Course review and discussion
25 Project presentations

Homework 8 due

Final project report due one week after final class

Fall 2018
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