Introduction to Small-World
Networks and Scale-Free Networks
Presented by Lillian Tseng
2005/11/3
OPLAB, NTUIM
1
Agenda








Introduction
Terminologies
Small-World Phenomenon
Small-World Network Model
Scale-Free Network Model
Comparisons
Application
Conclusion
2005/11/3
2
Introduction
2005/11/3
OPLAB, NTUIM
3
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!!
2005/11/3
4
2005/11/3
5
2005/11/3
6
2005/11/3
7
2005/11/3
8
2005/11/3
9
Categories of Complex Networks
Complex Networks
Social
Networks
Friendship
Sexual contact
Intermarriages
Business Relationships
Communication Records
Collaboration
(film actors)
(company directors)
(coauthor in academics)
(co-appearance)
2005/11/3
Technological
(Man-made)
Networks
Information
(Knowledge)
Networks
Biological
Networks
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
10
Terminologies
2005/11/3
OPLAB, NTUIM
11
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
2005/11/3
12
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
2005/11/3
13
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.
2005/11/3
14
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

2005/11/3
Definition II
15
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)
2005/11/3
16
Clustering Coefficient (C) (cont.)

Definition II (Watts and Strogatz proposed)

Example


2005/11/3
Definition I: C = 3/8
Definition II: C = 13/30
17
Small-World Phenomenon
2005/11/3
OPLAB, NTUIM
18
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…
2005/11/3
19
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
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.
2005/11/3
20
The Small World Experiment
(cont.)
2005/11/3
21
The Small World Experiment
(cont.)
Path Length
2005/11/3
Clustering Coefficient
22
“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.
2005/11/3
23
“Six Degrees of Separation” (cont.)

Duncan Watts et al. did it again via e-mails (384 out of
60,000) in 2003.
2005/11/3
24
2005/11/3
25
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
2005/11/3
26
Kevin Bacon & Harrison Ford
Top
Gun
Witness
A Few
Good
Men
Star Wars
2005/11/3
27
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.
2005/11/3
28
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.”
2005/11/3
29
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.
2005/11/3
30
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.
2005/11/3
31
Small-World Network Model
2005/11/3
OPLAB, NTUIM
32
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.
2005/11/3
33
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.
2005/11/3
34
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.
35
2005/11/3
Small-World Networks (cont.)
2005/11/3
36
Small-World Networks (cont.)
• Low mean path length
• High clustering coefficient
2005/11/3
37
Power Grid NW USA-Canada
|V| = 4,941
max = 19
aver = 2.67
L = 18.7 (12.4)
C = 0.08 (0.005)
2005/11/3
38
Scale-Free Network Model
2005/11/3
OPLAB, NTUIM
39
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.
2005/11/3
40
Degree Distribution
Poisson distribution
Exponential Network
2005/11/3
Power-law distribution
Scale-free Network
41
Degree distribution (cont.)

Continuous hierarchy of
vertices
 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
vertices
2005/11/3
42
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
<l>
nd.edu
2005/11/3
43
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
-out
P2005/11/3
(k)
~
k
out
Pin(k) ~ k- in
44
INTERNET BACKBONE
Nodes: computers, routers
Links: physical lines
2005/11/3
(Faloutsos, Faloutsos and Faloutsos, 1999)
45
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
2005/11/3
46
SCIENCE CITATION INDEX
Nodes: papers
Links: citations
25
Witten-Sander
PRL 1981
1736 PRL papers (1988)
2212
P(k) ~k-
( = 3)
2005/11/3
(S. Redner, 1998)47
SCIENCE COAUTHORSHIP
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.
2005/11/3
Liljeros et al. Nature 2001
49
Food Web
Nodes: trophic species
Links: trophic interactions
2005/11/3
R. Sole (cond-mat/0011195) 50
Metabolic Network
Nodes: chemicals (substrates)
Links: bio-chemical reactions
2005/11/3
51
Metabolic network
Archaea
Bacteria
Eukaryotes
Organisms from all three domains of life are
scale-free networks!
H. Jeong, B. Tombor,
52
2005/11/3 R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000)
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.
2005/11/3
53
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
j k j k -.
The result is a network with degree distribution 
P(k)
2005/11/3
54
Creation of Scale-Free Networks
2005/11/3
55
Small-World Networks v.s. ScaleFree Networks

Small-world networks
 Properties: mean path
length / clustering
coefficient
 Democratic
(homogeneous
vertices)
 Egalitarian (singlescale)
2005/11/3

Scale-free networks
 Property: degree
distribution
 Undemocratic
(heterogeneous
vertices)
 Aristocratic (scalefree)
 A subset of smallworld networks (?)
56
Single-Scale Networks
2005/11/3
Proc Nat Acad Sci USA 97, 11149 (2000)
57
Scale-Free Networks
Nature 411, 907 (2001)
2005/11/3
Phys Rev Lett 88, 138701 (2002) 58
Survivability of Small-World
Networks and Scale-Free Networks
d=the diameter of
the network
2005/11/3
59
Survivability of Small-World Networks and
Scale-Free Networks (cont.)
2005/11/3
60
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.
2005/11/3
61
Application
2005/11/3
OPLAB, NTUIM
62
Applications




Social search / Network navigation
Decision making
Mobile ad hoc networks
Peer-to-peer networks.
2005/11/3
63
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 Milgram’s experiment.
2005/11/3
64
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 < 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!
2005/11/3

65
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).
2005/11/3
66
Conclusion
2005/11/3
OPLAB, NTUIM
71
Conclusion

What’s small-world phenomenon
 Six degrees of separation
 Shortcuts
Networks with small-world
property
 Small-world networks



High clustering coefficient
Low mean path length
10
10
10
P(k)

10
10
10
0
-5
-10
-15
-20
-25
P(k) ~ Poisson
P(k) ~ k-3
Scale-free networks

2005/11/3
Power-law distribution
10
-30
10
0
10
1
10
2
10
3
k
72
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.
2005/11/3
73
Q&A
Thanks for your listening ^_^
2005/11/3
OPLAB, NTUIM
74