1.022 | Fall 2018 | Undergraduate

Introduction to Network Models



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.


Introduction to graph theory

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


Strong and weak ties, triadic closure, and homophily

Homework 1 distributed


Centrality measures

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


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.


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

Network models

  • Graphs as realizations of stochastic processes: Introduce the general idea.
  • Friendship paradox.
  • 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

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.


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


Linear dynamical systems

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

Homework 5 distributed during Lecture 14

Homework 4 due by Lecture 14

Markov chains

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


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

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.



  • 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

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 

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)

Application of game theory to networks

  • Traffic equilibrium: non-atomic traffic models.
  • Braess’s Paradox.
  • 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

Course Info

As Taught In
Fall 2018
Learning Resource Types
Problem Sets
Lecture Notes