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Agenda: Thursday, Feb 3

- Midterm date: Thursday, March 3
- New readings in Watts
- Our navigation experiment: some analysis
- Brief introduction to graph theory

News and Notes: Tuesday Feb 8

- From the Field: NY Times article 2/8 on hate groups on Orkut
- Duncan Watts talk Friday Feb 11 at noon!
- No MK office hours tomorrow
- Return of NW Construction, Task 1:
- first of all, staple your own work
- grading:
- 2/2: proceed as described
- 1/2: some problems, usually of specificity
- 0/2: fundamental flaw or lack of clarity
- if you received 2/2: leave your assignment here
- if you received 1/2: leave your assignment here, or revise and return on Thursday
- if you received 0/2: revise and return on Thursday
- Next Tuesday’s class:
- MK out of town, but mandatory class experiment
- once again, print and bring your Lifester neighbor profiles
- Today’s agenda:
- further analysis of Lifester NW navigation experiment
- quick review and completion of Intro to Graph Theory
- start on Social Network Theory

Description of the Experiment

- Participation is mandatory and for credit
- If you don’t have your Lifester neighbor profiles, you cannot participate
- unless you have memorized your neighbor info
- We will play two rounds
- In each round, each of you will be the source of a navigation chain
- You will be given a destination user to route a form to
- Give the form to one of your Lifester neighbors who you think is “closer” to the target
- Write your Lifester UserID on forms you receive, and continue to forward them towards their destinations
- Points will be deducted for violations of the neighborhood structure
- In one round, you will be given the Lifester profile of the destination
- In the other round, you will not be given the destination profile
- Then we’ll do some brief analysis with more detail to follow

With destination profile:

optimal mean = 3.67

class mean = 5.18

delta = 1.51

2 cycles

Without destination profile:

optimal mean = 3.6

class mean = 5.48

delta = 1.86

4 cycles

Undirected Graphs

- Recall our basic definitions:
- set of vertices denoted 1,…N; size of graph is N
- edge is an (unordered) pair (i,j)
- (i,j) is the same as (j,i)
- indicates that i and j are directly connected
- a graph G consists of the vertices and edges
- maximum number of edges: N(N-1)/2 (order N^2)
- i and j connected if there is a path of edges between them
- all-pairs shortest paths: efficient computation via Dijkstra's algorithm (another)
- Subgraph of G:
- restrict attention to certain vertices and edges between them
- Connected components of G:
- subgraphs determined by mutual connectivity
- connected graph: only one connected component
- complete graph: edge between all pairs of vertices

Complexity Theory in One Slide

N^3

N^2

computation time

computation time

polynomials:

tractable

linear functions:

tractable

size of graph

size of graph

2^N

computation time

exponential:

intractable

size of graph

- 1000^2 = 1 million
- 2^1000: not that many atoms!
- most known problems:
- either low-degree polynomial…
- … or exponential

Cliques and Independent Sets

- A clique in a graph G is a set of vertices:
- informal: that are all directly connected to each other
- formal: whose induced subgraph is complete
- all vertices in direct communication, exchange, competition, etc.
- the tightest possible “social structure”
- an edge is a clique of just 2 vertices
- generally interested in large cliques
- Independent set:
- set of vertices whose induced subgraph is empty (no edges)
- vertices entirely isolated from each other without help of others
- Maximum clique or independent set: largest in the graph
- Maximal clique or independent set: can’t grow any larger

Some Interesting Properties

- Computation of cliques and independent sets:
- maximal: easy, can just be greedy
- maximum: difficult --- believed to be intractable (NP-hard)
- computation time scales exponentially with graph size
- however, approximations are possible
- Social design and Ramsey theory:
- suppose large cliques or independent sets are viewed as “bad”
- e.g. in trade:
- large clique: too much collusion possible
- large independent set: impoverished subpopulation
- would be natural to want to find networks with neither
- Ramsey theory: may not be possible!
- Any graph with N vertices will have either a clique or an independent set of size > log(N)
- A nontrivial “accounting identity”; more later

Graph Colorings

- A coloring of an undirected graph is:
- an assignment of a color (label) to each vertex
- such that no pair connected by an edge have the same color
- chromatic number of graph G: fewest colors needed
- Example application:
- classes and exam slots
- chromatic number determines length of exam period
- Here’s a coloring demo
- Computation of chromatic numbers is hard
- (poor) approximations are possible
- Interesting fact: the four-color theorem for planar graphs

Matchings in Graphs

- A matching of an undirected graph is:
- a subset of the edges
- such that no vertex is “touched” more than once
- perfect matching: every vertex touched exactly once
- perfect matchings may not always exist (e.g. N odd)
- maximum matching: largest number of edges
- Can be found efficiently; here is a perfect matching demo
- Example applications:
- pairing of compatible partners
- perfect matching: nobody “left out”
- jobs and qualified workers
- perfect matching: full employment, and all jobs filled
- clients and servers
- perfect matching: all clients served, and no server idle

Cuts in Graphs

- A cut of a (connected) undirected graph is:
- a subset of the edges (edge cut) or vertices (vertex cut)
- such that the removal of this set would disconnect the graph
- min/maximum cut: smallest/largest (minimal) number
- computation can be done efficiently
- Often related to robustness of the network
- small cuts ~ vulnerability
- edge cut: failure of links
- vertex cut: failure of “individuals”
- random versus maliciously chosen failures (terrorism)

Spanning Trees

- A spanning treeof a (connected) undirected graph is:
- a subgraph G’ of the original graph G
- such that G’ is connected but has no cycles (a tree)
- minimum spanning tree: fewest edges
- computation: can be done efficiently
- Minimal subgraphs needed for complete communication
- Different spanning tree provide different solutions
- Applications:
- minimizing wire usage in circuit design

Directed Graphs

- Graphs in which the edges have a direction
- Edge (u,v) means u v; may also have (v,u)
- Common for capturing asymmetric relations
- Common examples:
- the web
- reporting/subordinate relationships
- corporate org charts
- code block diagrams
- causality diagrams

Weighted Graphs

- Each edge/vertex annotated by a weight or capacity
- Directed or undirected
- Used to model
- cost of transmission, latency
- capacity of link
- hubs and authorities (Google PageRank algorithm)
- Common problem: network flow, efficiently solvable

Planar Graphs

- Graphs which can be drawn in the plane with no edges crossing (except at vertices)
- Of interest for
- maps of the physical world
- circuit/VLSI design
- data visualization
- Graphs of higher genus
- Planarity testing efficiently solvable

Bipartite Graphs

- Vertices divided into two sets
- Edges only between the two sets
- Example: affiliation networks
- vertices are individuals and organizations
- edge if an individual belongs to an organization
- Men and women, servers and clients, jobs and workers
- Some problems easier to compute on bipartite graphs

We’ll make use of these graph types… but will

generally be looking at classes of graphs generated

according to a probability distribution, rather than

obeying some fixed set of deterministic properties.

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