Agenda: Thursday, Feb 3

1
Agenda Thursday, Feb 3

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

2
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 Tuesdays class
• MK out of town, but mandatory class experiment
• once again, print and bring your Lifester
  neighbor profiles
• Todays agenda
• further analysis of Lifester NW navigation
  experiment
• quick review and completion of Intro to Graph
  Theory
• start on Social Network Theory

3
Description of the Experiment

• Participation is mandatory and for credit
• If you dont 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 well do some brief analysis with more
  detail to follow

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diameter worst-case 5 average 2.86
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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
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Comparison to Random Walks
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degree vs. betweenness, class chains
number of chains
degree of user
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degree vs. betweenness, optimal chains
number of chains
degree of user
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A Brief Introduction to Graph Theory

• Networked Life
• CSE 112
• Spring 2005
• Prof. Michael Kearns

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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 N2)
• 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

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Complexity Theory in One Slide

• 10002 1 million
• 21000 not that many atoms!
• most known problems
• either low-degree polynomial
• or exponential

12
Properties and Measuresof Graphs
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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 cant grow any
  larger

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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

15
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
• Heres a coloring demo
• Computation of chromatic numbers is hard
• (poor) approximations are possible
• Interesting fact the four-color theorem for
  planar graphs

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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

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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)

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Spanning Trees

• A spanning tree of 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

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Summary of Graph Properties
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Special Types of Graphs
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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

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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

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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

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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

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Well 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.