Introduction to Small-World Networks and Scale-Free Networks by iuq51574

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									           Introduction to Small-World
        Networks and Scale-Free Networks

              Presented by Lillian Tseng

2005/11/3             OPLAB, NTUIM         1

   Introduction
   Terminologies
   Small-World Phenomenon
   Small-World Network Model
   Scale-Free Network Model
   Comparisons
   Application
   Conclusion
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Why is Network Interesting?

   Lots of important problems can be represented as
   Any system comprising many individuals
    between which some relation can be defined can
    be mapped as a network.
   Interactions between individuals make the
    network complex.
   Networks are ubiquitous!!

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         Categories of Complex Networks
                                   Complex Networks

                           Technological           Information
       Social                                                            Biological
                           (Man-made)             (Knowledge)
      Networks                                                           Networks
                             Networks               Networks

       Friendship                Internet
     Sexual contact         Software classes
                                                      WWW             Metabolic pathways
     Intermarriages           Airline routes
                                                       P2P            Protein interactions
 Business Relationships      Railway routes
                                                 Academic citations   Genetic regulatory
Communication Records          Roadways
                                                  Patent citations           Neural
      Collaboration            Telephone
                                                   Word classes          Blood vessels
      (film actors)              Delivery
                                                    Preference             Food web
  (company directors)     Electric power grids
(coauthor in academics)     Electronic circuit
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Vertex and Edge
   Vertex (pl. Vertices)
      Node (computer science), Site (physics),
       Actor (sociology)
   Edge
      Link (computer science), Bond (physics), Tie
      Directed: citations
      Undirected: committee membership
      Weighted: friendship

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Degree and Component
   Degree
     The number of edges connected to a vertex.
     In-degree / Out-degree in a directed graph
   Component
     Set of vertices to be reached from a vertex by
       paths running along edges.
     In-component / Out-component in a directed
     Giant component

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Diameter (d)

   Geodesic path (Shortest path)
     The shortest path from one vertex to another.
   Geodesic path length / Shortest path length /
   Diameter (in number of edges)
     The longest geodesic path length between any
        two vertices.

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Mean Path Length (L)

   Mean (geodesic) path length L – global property
     The shortest path between two vertices,
       averaged over all pairs of vertices.
     Definition I

           Definition II

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Clustering Coefficient (C)

   Clustering coefficient C –local property
     The mean probability that two vertices that
       are network neighbors of the same other
       vertex will themselves be neighbors.
     Definition I (fraction of transitive triples,
       widely used in the sociology literature)

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Clustering Coefficient (C) (cont.)

           Definition II (Watts and Strogatz proposed)

           Example
               Definition I: C = 3/8
               Definition II: C = 13/30

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     Small-World Phenomenon

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The Small World Problem / Effect
   First mentioned in a short story in 1929 by Hungarian
    writer Frigyes Karinthy.
   30 years later, became a research problem “contact and
      In 1958, Pool and Kochen asked “what is the
         probability that two strangers will have a mutual
         friend?” (What is the structure of social networks?)
           i.e. the small world of cocktail parties
      Then asked a harder question: “What about when
         there is no mutual friend --- how long would the
         chain of intermediaries be?”
      Too hard…
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The Small World Experiment
   In 1967, Stanley Milgram (and his student Jeffrey Travers)
    designed an experiment based on Pool and Kochen’s
    work. (How many intermediaries are needed to move a
    letter from person A to person B through a chain of
      A single target in Boston.
      300 initial senders in Boston (100) and Omaha (in
         Nebraska) (200).
      Each sender was asked to forward a packet to a
         friend who was closer to the target.
      The friends got the same instructions.

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The Small World Experiment

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The Small World Experiment
            Path Length   Clustering Coefficient

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“Six Degrees of Separation”
   Travers and Milgrams’ protocol generated 300 letter
    chains of which 44 (?) reached the target.
   Found that typical chain length was 6.
      “What a small-world!!”
   Led to the famous phrase: “Six Degrees of Separation.”

   Then not much happened for another 30 years.
      Theory was too hard to do with pencil and paper.
      Data was too hard to collect manually.

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“Six Degrees of Separation” (cont.)
      Duncan Watts et al. did it again via e-mails (384 out of
       60,000) in 2003.

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Six Degrees of Bacon
   Kevin Bacon has acted creditedly in 56 movies so far
      Any body who has acted in a film with Bacon has a
         bacon number of 1.
      Anybody who does not have a bacon number 1 but
         has worked with somebody who does, they have
         bacon number 2, and so on.
   Most people in American movies have a number 4 or less.
    Given that there are about 630,000 such people, and this
    is remarkable.
   The Oracle of Bacon

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Kevin Bacon & Harrison Ford

        A Few

                      Star Wars

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What is “Six Degree”?
   “Six degrees of separation between us and everyone else
    on this planet.”
      A play : John Guare, 1990.
   An urban myth? (“Six handshakes to the President”)

   The Weak Version
      There exists a short path from anybody to anybody
   The Strong Version
      There is a path that can be found using local
        information only.
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The Caveman World

   Many caves, and people know only others in their
    caves, and know all of them.
   Clearly, there is no way to get a letter across to
    somebody in another cave.
   If we change things so that the head-person of a
    cave is likely to know other head-people, letters
    might be got across, but still slowly.
   There is too much “acquaintance-overlap.”

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The World of Chatting

   People meet others over the net.
   In these over-the-net-only interactions, there is
    almost no common friends.
   Again, if a message needed to be sent across, it
    would be hard to figure out how to route it.

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Small Worlds Are Between These
   When there is some, but not very high, overlap
    between acquaintances of two people who are
    acquainted, small worlds results.
   If somebody knows people in different groups
    (caves?), they can act as linchpins that connect
    the small world.
   For example, cognitive scientists are lynchpins
    that connect philosophers, linguists, computer
    scientists etc.
   Bruce Lee is a linchpin who connects Hollywood
    to its Chinese counterpart.
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     Small-World Network Model

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The “New” Science of Networks
   Mid 90’s, Duncan Watts and Steve Strogatz
    worked on another problem altogether.
   Decided to think about the urban myth.
   They had three advantages.
      They did not know anything.
      They had many faster computers.
      Their background in physics and mathematics
       caused them to think about the problem
       somewhat differently.

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The “New” Science of Networks

   Instead of asking “How small is the actual
    world?”, they asked “What would it take for any
    world at all to be small?”
   As it turned out, the answer was not much.
      Some source of “order” and “regularity”
      The tiniest amount of “randomness”
   Small World Networks should be everywhere.

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Small-World Networks
 high clustering    high clustering      low clustering
 high distance      low distance         low distance

• fraction p of the links is converted into shortcuts.
• Randomly rewire each edge with probability p to introduce
increased amount of disorder.
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Small-World Networks (cont.)

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Small-World Networks (cont.)

                 • Low mean path length
                 • High clustering coefficient

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Power Grid NW USA-Canada

                 |V| = 4,941
                 max = 19
                 aver = 2.67
                 L = 18.7 (12.4)
                 C = 0.08 (0.005)

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     Scale-Free Network Model

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What is Scale-Free?

   The term “scale-free” refers to any distribution
    functional form f(x) that remains unchanged to
    within a multiplicative factor under a rescaling of
    the independent variable x.
   In effect, this means power-law forms f(x) =x-,
    since these are the only solutions to f(ax) = bf(x),
    and hence “power-law” and “scale-free” are, for
    some purposes, synonymous.

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Degree Distribution
  Poisson distribution   Power-law distribution

  Exponential Network
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Degree distribution (cont.)
   Continuous hierarchy of
      Smooth transition
         from biggest hub over
         several more slightly
         less big hubs to even
         more even smaller
         vertices…down to the
         huge mass of tiny

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    World Wide Web
           Nodes: WWW documents
           Links: URL links
           Based on 800 million web pages
 Finite size scaling: create a network with N nodes with Pin(k) and Pout(k)

                              < l > = 0.35 + 2.06 log(N)
                                                 19 degrees of separation


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What did we expect?
                                        k ~ 6
                                  P(k=500) ~ 10-99
                                  NWWW ~ 109
                                   N(k=500)~10-90
In fact, we find:

         out= 2.45         in = 2.1
                                          P(k=500) ~ 10-6
                                          NWWW ~ 109
                                           N(k=500) ~ 103

   Pout(k) ~ k-out
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            Nodes: computers, routers
            Links: physical lines

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                            (Faloutsos, Faloutsos and Faloutsos, 1999)
       Nodes: actors
       Links: cast jointly

                   Days of Thunder (1990)
                   Far and Away    (1992)
                   Eyes Wide Shut (1999)

   N = 212,250 actors
   k = 28.78

   P(k) ~k-
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       Nodes: papers
       Links: citations
                               PRL 1981
     1736 PRL papers (1988)

                              P(k) ~k-
                              ( = 3)

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                                          (S. Redner, 1998)
            Nodes: scientist (authors)
            Links: write paper together

2005/11/3                     (Newman, 2000, H. Jeong et al 2001)48
            SEX WEB
                 Nodes: people (females, males)
                 Links: sexual relationships

                      4781 Swedes; 18-74;
                      59% response rate.

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                           Liljeros et al. Nature 2001
                   Food Web
            Nodes: trophic species
            Links: trophic interactions

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                 Metabolic Network
Nodes: chemicals (substrates)

Links: bio-chemical reactions

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                       Metabolic network

                   Archaea               Bacteria               Eukaryotes

                Organisms from all three domains of life are
                          scale-free networks!

H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000)
Characteristics of Scale-Free

   The number of vertices N is not fixed.
      Networks continuously expand by the
       addition of new vertices.
   The attachment is not uniform.
      A vertex is linked with higher probability to a
       vertex that already has a large number of

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Characteristics of Scale-Free
Networks (cont.)
   Growth
      Start with few linked-up vertices and, at each time
        step, a new vertex with m edges is added.
      Potential for imbalance
   Preferential Attachment
      Each edge connects with a vertex in the network
        according to a probability i proportional to the
        connectivity ki of the vertex.
      Emergence of hubs                              k
                                             (ki )      i

   The result is a network with degree distribution  j k j k -.

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Creation of Scale-Free Networks

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Small-World Networks v.s. Scale-
Free Networks

   Small-world networks          Scale-free networks
      Properties: mean path         Property: degree
       length / clustering             distribution
       coefficient                   Undemocratic
      Democratic                      (heterogeneous
       (homogeneous                    vertices)
       vertices)                     Aristocratic (scale-
      Egalitarian (single-            free)
       scale)                        A subset of small-
                                       world networks (?)

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Single-Scale Networks

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Scale-Free Networks

                Nature 411, 907 (2001)
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Survivability of Small-World
Networks and Scale-Free Networks

                         d=the diameter of
                            the network

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Survivability of Small-World Networks and
Scale-Free Networks (cont.)

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Short Summary
   The numerical simulations indicate
      There is a strong correlation between
       robustness and network topology.
      Scale-free networks are more robust than
       random networks against random vertex
       failures (error tolerance) because of their
       heterogeneous topology, but are more fragile
       when the most connected vertices are targeted
       (attack vulnerability / low attack survivability)
       with the same reason.
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   Social search / Network navigation
   Decision making
   Mobile ad hoc networks
   Peer-to-peer networks.

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Social Search
   Find jobs.
      We tend to use “weak ties” (Granovetter) and also
          “friends of friends”.
   It is true that at any point in time, someone who is six
    degrees away is probably impossible to find and would
    not help you if you could find them.
   But, social networks are not static, and they can be altered
   Over time, we can navigate out to six degrees.
   Search process is just like Milgram’s experiment.

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Social Search (Experiment)
   Identical protocol to Travers and Milgrams’, but
    conducted via the Internet.
 60,000 participants from 170 countries attempting to
    reach 18 different targets
 Important results:
       Median true chain length 5 < L < 7.
       Geography and Occupation most important.
       Weak ties help, but medium-strength ties typical.
       Professional ties lead to success.
       Hubs don’t seem to matter.
       Participation and Perception matter most!
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Collective Problem Solving
   Small-world problem is an example of “social search.”
      Individuals search for remote targets by forwarding
         message to acquaintance.
      Social networks turn out to be searchable.
      But search process is collective in that chain knows
         more about the network than any individual.
      Not possible in all networks.
   Social search is relevant not only to finding jobs and
    locating answers / resources (i.e. individual problem
    solving) but also collective problem solving (innovation /
    recovery from catastrophe).

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   What’s small-world phenomenon                        10

      Six degrees of separation
      Shortcuts
   Networks with small-world                            10



      Small-world networks
                   High clustering coefficient
                   Low mean path length                 10
                                                                        P(k) ~ Poisson
                                                                        P(k) ~ k
         Scale-free networks                                 -30
                   Power-law distribution                     10


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Conclusion (cont.)

    All complex networks in nature seems to have
     power-law degree distribution.
       It is far from being the case!!
    Some networks have degree distribution with
     exponential tail.
       They do not belong to random graph because
         of evolving property.
    Evolving networks can have both power-law and
     exponential degree distributions.
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            Thanks for your listening ^_^

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