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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!

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agenda thursday feb 3
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
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
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
slide4
diameter:

worst-case: 5

average: 2.86

slide5
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

slide7
degree vs. betweenness, class chains

number of chains

degree of user

slide8
degree vs. betweenness, optimal chains

number of chains

degree of user

a brief introduction to graph theory

A Brief Introduction to Graph Theory

Networked Life

CSE 112

Spring 2005

Prof. Michael Kearns

undirected graphs
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
slide11
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
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
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
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
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
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
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
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
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
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
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
slide25
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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