Introduction to Small-World Networks and Scale-Free Networks

1
Introduction to Small-World Networks and
Scale-Free Networks

• Presented by Lillian Tseng

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Agenda

• Introduction
• Terminologies
• Small-World Phenomenon
• Small-World Network Model
• Scale-Free Network Model
• Comparisons
• Application
• Conclusion

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Introduction
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Why is Network Interesting?

• Lots of important problems can be represented as
  networks.
• 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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Internet-Map
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Categories of Complex Networks
Complex Networks
Social Networks
Technological(Man-made) Networks
Information (Knowledge) Networks
Biological Networks
Friendship Sexual contact Intermarriages Business
Relationships Communication Records Collaboration
(film actors) (company directors) (coauthor in
academics) (co-appearance)
Internet Software classes Airline routes Railway
routes Roadways Telephone Delivery Electric power
grids Electronic circuit
WWW P2P Academic citations Patent citations Word
classes Preference
Metabolic pathways Protein interactions Genetic
regulatory Neural Blood vessels Food web
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Terminologies
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Vertex and Edge

• Vertex (pl. Vertices)
• Node (computer science), Site (physics), Actor
  (sociology)
• Edge
• Link (computer science), Bond (physics), Tie
  (sociology)
• 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 graph
• 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 /
  Distance
• 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 influence.
• 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
  Kochens work. (How many intermediaries are
  needed to move a letter from person A to person B
  through a chain of acquaintances?)
• 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 (cont.)
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The Small World Experiment (cont.)
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
• http//www.cs.virginia.edu/oracle

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Kevin Bacon Harrison Ford
Top Gun
Witness
A Few Good Men
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
  else.
• 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 Extremes

• 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 90s, 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 (cont.)

• 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 distance
high clustering low distance
low clustering 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
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Degree distribution (cont.)

• Continuous hierarchy of vertices
• Smooth transition from biggest hub over several
  more slightly less big hubs to even more even
  smaller verticesdown to the huge mass of tiny
  vertices

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World Wide Web
Nodes WWW documents Links URL links
Based on 800 million web pages
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What did we expect?
?k? 6 P(k500) 10-99
NWWW 109 ? N(k500)10-90
In fact, we find
?out 2.45
? in 2.1
P(k500) 10-6
NWWW 109 ? N(k500) 103
Pout(k) k-?out
Pin(k) k- ?in
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INTERNET BACKBONE
Nodes computers, routers Links physical lines
(Faloutsos, Faloutsos and Faloutsos, 1999)
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ACTOR CONNECTIVITIES
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-?
?2.3
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SCIENCE CITATION INDEX
Nodes papers Links citations
Witten-Sander PRL 1981
1736 PRL papers (1988)
P(k) k-?
(? 3)
(S. Redner, 1998)
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SCIENCE COAUTHORSHIP
Nodes scientist (authors) Links write paper
together
(Newman, 2000, H. Jeong et al 2001)
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SEX WEB
Nodes people (females, males) Links sexual
relationships
4781 Swedes 18-74 59 response rate.
Liljeros et al. Nature 2001
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Food Web
Nodes trophic species Links trophic
interactions
R. Sole (cond-mat/0011195)
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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)
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Characteristics of Scale-Free Networks

• 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 edges.

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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
• The result is a network with degree distribution
  P(k) ? k -?.

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

• Scale-free networks
• Property degree distribution
• Undemocratic (heterogeneous vertices)
• Aristocratic (scale-free)
• A subset of small-world networks (?)

• Small-world networks
• Properties mean path length / clustering
  coefficient
• Democratic (homogeneous vertices)
• Egalitarian (single-scale)

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Single-Scale Networks
Proc Nat Acad Sci USA 97, 11149 (2000)
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Scale-Free Networks
Nature 411, 907 (2001) Phys Rev Lett 88, 138701
(2002)
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Survivability of Small-World Networks and
Scale-Free Networks
dthe 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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Application
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Applications

• 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 strategically.
• Over time, we can navigate out to six degrees.
• Search process is just like Milgrams experiment.

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Social Search (Experiment)

• Identical protocol to Travers and Milgrams, but
  conducted via the Internet.
• http//smallworld.sociology.columbia.edu
• 60,000 participants from 170 countries attempting
  to reach 18 different targets
• Important results
• Median true chain length 5 lt L lt 7.
• Geography and Occupation most important.
• Weak ties help, but medium-strength ties typical.
• Professional ties lead to success.
• Hubs dont 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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Network Navigation

• Two fundamental components in the small-world
• Short chains are ubiquitous.
• Individuals operating with purely local
  information are easy at finding these chains.

• Agents are on a grid.
• Everybody is connected to their neighbors.
• But they are also connected to k other agents
  randomly.

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Network Navigation (cont.)

• The distribution could be uniform, or biased
  towards closer agents.
• It could be inversely proportional to the
  distance d from us to that agent, or inversely
  proportional to the square of the distance (sort
  of like gravitation).
• These can be represented as inversely
  proportional to d to the power r (clustering
  exponent), where r is 0, 1, 2 or above.

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Network Navigation (cont.)

• If r is 0, the neighbor is chosen randomly, and
  the world is like the solarium world.
• If r is very high, you only know your immediate
  neighbors and the world is like the caveman
  world.
• For intermediate values, we get more and more
  small-world-like behavior.
• There is always a findable path whose length is
  not too big only when r is 2!!

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Network Navigation (cont.)

• For any other R (smaller or bigger than 2), the
  expected length of a findable path is larger.
• Efficient navigability is a fundamental property
  of only some small-world structures (???).
• The correlation between local structure and
  long-range connections provides critical cues for
  finding paths through networks.

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Conclusion
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Conclusion

• Whats small-world phenomenon
• Six degrees of separation
• Shortcuts
• Networks with small-world property
• Small-world networks
• High clustering coefficient
• Low mean path length
• Scale-free networks
• Power-law distribution

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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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Q A

• Thanks for your listening _

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References (Papers)

• S. Milgram, The small world problem, Psych.
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• D. J. Watts and S. H. Strogatz, Collective
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  (1998), pp. 440442.
• M. Barthelemy and L. A. N. Amaral,
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• S. Lawrence and C. L. Giles, Accessibility of
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• R. Albert, H. Jeong, and A.-L. Barabasi,
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• A.-L. Barabasi and R. Albert, Emergence of
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• A.-L. Barabasi, R. Albert, and H. Jeong,
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• J. M. Kleinberg, Navigation in a small world,
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References (Surveys)

• R. Albertand A.-L. Barabasi, Statistical
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• S. N. Dorogovtsev and J. F. F. Mendes,
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  literature, and devote the larger part of
  attention to the models of growing network
  models.
• M. E. J. Newman, The structure and function of
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• S. H. Strogatz, Exploring complex networks,
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• Short Survey (the above two) devote discussion
  on the small-world model.

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References (Books)

• A.-L. Barabasi, Linked The New Science of
  Networks, Perseus, Cambridge, MA, 2002.
• Focus on scale-free network
• M. Buchanan, Nexus Small Worlds and the
  Groundbreaking Science of Networks, Norton, New
  York, 2002.
• From the point of view of a science journalist
• D. J. Watts, Six Degrees The Science of a
  Connected Age, Norton, New York, 2003.
• From the point of view of a sociologist

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Reference

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Reference (cont.)

• J. O. Kephartand S. R. White, Directed-graph
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Reference (cont.)

• For food webs
• J. A. Dunne, R. J. Williams, and N. D.
  Martinez, Food-webstructur e and network theory
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• For metabolic networks
• H. Jeong, S. Mason, A.-L. Barabasi, and Z. N.
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