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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 l.jpg
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 l.jpg
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


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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


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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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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 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


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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



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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: can’t 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


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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


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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 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




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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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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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