1

Course specifics, motivation, and intro to graph theory
 Course specifics: 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, smallworld phenomenon, adjacency and incidence matrices.


3

Strong and weak ties, triadic closure, and homophily
Homework 1 distributed


4

Centrality measures
 Detection and identication 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.
 Friendship paradox.
 ErdősRényi graphs, branching processes. Denition, examples, phase transition, connectivity, diameter, and giant component.
Homework 3 distributed during Lecture 10

Homework 2 due by Lecture 9

11

Configuration model and smallworld graphs
 Conguration model, emergence of the giant component.
 Smallworld graphs: Denition 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 field 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
 PerronFrobenius theorem.
 Random walk on graphs.


16–17

Information spread and distributed computation
 Conductance and information spread.
 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: Decisionmaking with many agents, utility maximization.
 Basic ingredients of a game: Strategic or normal form games.
 Strategies: Finite / Infinite 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 writeup of final project (nondefault version) due

22

Introduction to game theory II
 Nash equilibrium: more examples.
 Multiplicity of equilibria.
 ParetoOptimality 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 infinite 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: nonatomic traffic models.
 Braess’s Paradox.
 SociallyOptimal 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
