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. 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
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	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: 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. 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