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Network theory
1. An Introduction to
Network Theory
Kyle Findlay
Kyle.Findlay@tns-global.com
R&D Executive
TNS Global Brand Equity Centre
Presented at the
SAMRA 2010 Conference
Mount Grace Country House and Spa, Magaliesburg, South Africa, from 2| to 5 June 2010
An Introduction to Network Theory | Kyle Findlay SAMRA 2010
2. An agent/object's actions are affected
by the actions of others around it.
What is a network?
Actions, choices, etc. are not made in isolation
i.e. they are contingent on others' actions, choices, etc.
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
3. “A collection of objects connected to
each other in some fashion”
[Watts, 2002]
4. Social groups Diseases Stem cells
The internet Neural networks (computer & human) Other cells
Cities Proteins & genes Plants
Quaking Aspen (one of the largest organisms in the world –
these trees represent a single organism with a shared root
system)
The blogosphere
Source: Six Degrees, Duncan Watts, 2002
Proliferation of landlines in London
What is a network?
Human genome
Rabbit cell
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
5. ▫ New paradigm: “real networks represent populations of individual components
that are actually doing something” [Watts, 2002]
In other words, networks are dynamic objects that are continually changing
Understanding a network is important because its structure affects the individual
components’ behaviour and the behaviour of the system as a whole
Networks used to be thought of as systems… structures
▫ Networks are key to understanding non-linear, dynamic
fixed
…just like those represented by almost every facet of the universe…
…from the atomic level right through to the cosmic level
▫ The important part is that the components are not acting independently – they
are affected by the components around them!
▫ Note: links between component may be physical (e.g. power cable, magnetism)
or conceptual (e.g. social connections)
What is a network? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
6. CAUTION: Gratuitous network shots
Data networks Air traffic network
Telecommunications networks Shipping (sea) networks
Source: Britain From Above (http://www.bbc.co.uk/britainfromabove) An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
7. Network thinking can be applied
almost anywhere!
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
8. URL: http://www.youtube.com/watch?v=PufTeIBNRJ4
Epidemiology (i.e. spread of diseases)
e.g. spread of foot & mouth disease in the UK in 2001 over 75 days
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
9. URL: http://www.youtube.com/watch?v=8C_dnP2fvxk
Physics
e.g. particle interactions, the structure of the universe
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010
10. URL: http://www.youtube.com/watch?v=lRZ2iEHFgGo URL: http://www.youtube.com/watch?v=AEoP-XtJueo
Engineering
e.g. creation of robust infrastructure (e.g. electricity, telecoms), rust formation (natural growth
processes similar to diffusion limited aggregation)
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
11. Vid not working
URL: http://www.youtube.com/watch?v=l-RoDv7c5ok URL: http://www.youtube.com/watch?v=o4g930pm8Ms
Technology
e.g. mapping the online world, making networks resilient in the face of cyber-terrorism,
optimising cellular networks, controlling air traffic
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
12. URL: http://www.youtube.com/watch?v=YadP3w7vkJA URL: http://www.youtube.com/watch?v=Sp8tLPDMUyg
Biology
e.g. fish swimming in schools, ant colonies, birds flying in formation, crickets chirping in
unison, giant honeybees shimmering
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
13. Source: The Human Brain Book by Rita Carter
Medicine
e.g. cell formation, nervous system, neural networks
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
14. URL: http://www.youtube.com/watch?v=9n9irapdON4 URL: http://www.youtube.com/watch?v=sD2yosZ9qDw
And, most interestingly…society
e.g. interactions between people (e.g. Facebook; group behaviour)
Where’s it applied? An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
15. Some terminology…
▫ Node = individual
components of a
network e.g.
people, power
stations, neurons,
etc.
▫ Edge = direct link
between
components
(referred to as a
dyad in context of
social networking
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
16. No connections Some nodes All relevant nodes
between nodes connected connected
c=0 c = 1/3 c=1
▫ Tells you how likely a node is to be connected to its neighbours…
…and, importantly, how likely that its neighbours are connected to each other
▫ Put another way, it tells you how close a node and its neighbors are to being a clique where
“everybody knows everyone else”
Important network features:
Clustering co-efficient
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
17. “Unclustered” network “Clustered” network
None of Ego’s friends know each other* All of Ego’s friends know each other
Important network features:
Clustering co-efficient
*Source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
18. ▫ A real-world example: CEOs of Fortune 500 companies
Which companies share directors? Clusters are
colour-coded
Important network features:
Clustering co-efficient
Source: http://flickr.com/photos/11242012@N07/1363558436 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
19. ▫ Average path length = the average number of
‘hops’ required to reach any other node in the
network
▫ “Six degrees of separation” average path length =
6
Important network features:
Average path length
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
20. ▫ The degree of a node is the number of connections
(or edges) it has coming in from, and going out to,
other nodes 1
2
10
3
9
Node 4
8
7
5
10 connections 6
or “edges”
Important network features:
Degree distribution
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
21. 3 main types of networks
1. Grid/lattice network 2. Small-world network 3. Random network
(structure, order) (a mix of order and randomness) (randomness)
β=0 << Level of randomness of links >> β=1
They sit on a continuum based on a few factors:
1
Randomness 2
Clustering 3
Ave path length
*Source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
22. 3 main types of networks:
Grid/Lattice network
▫ Simplest form of network with nodes ranged
geometrically
▫ Low degree (nodes only connected to closest
neighbours)
▫ High clustering
▫ Long average path length (no shortcuts – have to
go through all nodes)
▫ Pros: methodical, easy to visualise
▫ Cons: not very good at modeling most real-
1D lattice
world networks
Molecule Diamond (crystal) lattice Bismuth crystal
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
23. 3 main types of networks:
Small world network
▫ Most nodes aren’t neighbours, but they can be
reached
from every other node by a small number of
hops or steps
Higher clustering co-efficient than to a few random
i.e. small average path length dueone would expect if
connections were made by pure random chance
re-wirings
− “A small world network, where nodes represent people and edges
connect people that know each other, captures the small world
phenomenon of strangers being linked by a mutual acquaintance”
Common in nature, including everything from the internet
to gene regulatory networks to ecosystems
Source: http://en.wikipedia.org/wiki/Small-world_network An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
24. 3 main types of networks:
Random network
▫ Lower clustering than small-world
networks generally
▫ No “force” or “bias” influencing how
links are created between nodes
i.e. probability of creating an edge/link is
independent of previous connections
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
25. The big picture
Nature
▫ Networks are evident
everywhere in nature
▫ In fact, most natural growth
Natural growth = evolutionary, iterative growth, where future growth is
constrained by previous growth patterns (referred to as path dependence)
processes come about due to
− i.e. growth follows the network structure
network behaviour
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
26. And market research?
▫ Networks better reflect reality and capture complexity i.e. non-linear dynamics
▫ Network theory helps us to better understand:
?How will word of my
brand permeate through
? How will negative publicity
about my brand spread and
? Who are the gatekeepers
in a community that
my target market? be interpreted? most affect the
flow of information?
?How is the market likely to
fall out in terms of
? What will the non-linear
market share impact be of a specific
(Double Jeopardy)? change in the market
e.g. change in market share,
perceptions, etc.
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
27. And market research?
▫ Network theory has been used to understand imagery and market barriers
Adjusting attributes and seeing knock-on effect in network
Using agent-based modeling to model this effect
Useful for word-of-mouth/viral approaches
− Watts and Peretti use network theory to
increase reach of WOM campaigns
Helps us avoid thinking about things in a vacuum
as it takes account of inter-related variables…
− … and provides us with counter-intuitive outcomes
that we may not have reached on our own
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
29. ▫ This is a very simple 2D
representation of how I
roughly visualise
information
propagating through a
network
It is very simple and
doesn’t take into
account many
concepts
But it is a visual aid
that helps one to start
thinking about interesting bits:
Some how
A network in action
information might
spread from person to
Source: http://www.funny-games.biz/reaction-effect.html
person
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
30. What does a highly “spreadable” idea look like?
• 1,636,967 views in two months (as at 25 April 2010)
• Performed in front of 75,000 people at Coachella Music Festival (California, April 2010)
URL: http://www.youtube.com/watch?v=Q77YBmtd2Rw
Some interesting bits:
How ideas spread
31. Who spreads ideas?
− Watts vs. Gladwell
vs.
− Mavens/influencers vs. forest fire
− Self-organised criticality
− K-shell decomposition?
Some interesting bits:
How ideas spread
32. Which ideas spread?
− Unpredictable
− Ideas that “fit”
Some interesting bits:
How ideas spread
33. ▫ Refers to systems in which many individual agents with
limited intelligence and information are able to pool
resources to accomplish a goal beyond the capabilities
of the individuals… while no single ant knows how toself-interest
− e.g.
only focused on build an ant colony
− e.g. in mind
without the bigger picturethe internet has grown organically over time
with no single person directing its growth
−
i.e. no grand designer
This is known as self-organisation and/or emergence, and is a property of
complex networks and non-linear, dynamic systems
Some interesting bits:
Distributed intelligence
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
34. ▫ Existence of such behaviour in organisms has many
implications for social, military and management
applications and is one of the most active areas of
research today!
Some interesting bits:
Distributed intelligence diffusion, memes,
Works best in small-world networks
Implications for knowledge
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
35. URL: http://www.youtube.com/watch?v=ozkBd2p2piU
Ant colony
Some interesting bits:
Distributed intelligence
Source: http://www.youtube.com/watch?v=ozkBd2p2piU An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
36. ▫ “On average, the first 5 random re-wirings
reduce the average path length of the network by
one-half, regardless of the size of the network”
[Watts, 2002]*
Random re-wirings
“8” “3”
Long average path length Dramatically reduced average path length
Some interesting bits:
Random re-wirings
*Source: Six Degrees, Duncan Watts, 2002, p.89 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
37. Some interesting bits:
Triadic closure
A B
C
▫ People are more likely to become acquainted over time when they have something in
common
i.e. we have a bias towards the familiar, thus reducing the pure randomness of
connections
Known as “homophily” - “birds of a feather flock together”
▫ Network connections don’t arise independently of each other…
…they are influenced by previous connections
▫ If A knows B…
…and B knows C…
…then A is much more likely to know C
*Source: Six Degrees, Duncan Watts, 2002, p.58-61 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
38. Some interesting bits:
Triadic closure
A B
C
This is why random re-wirings are so effective at reducing the ave. path length…
− …they help connect clusters, or ‘cliques’, that might otherwise exist in isolation
This is the strength of the small-world network:
− High clustering and a relatively small amount of random re-wirings allows for a
dramatically reduced average path length…
− …allowing everyone to connect to everyone else in relatively few steps e.g. “six degrees
of separation”
*Source: Six Degrees, Duncan Watts, 2002, p.58-61 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
39. Some interesting bits:
Triadic closure
•This “birds of a feather flock together” effect was modeled by Watts and Strogatz*
• They used α (alpha) to represent level of preference to only connect with friends of
friends
• Low α = strong preference to only connect with friends of friends (triadic closures occur,
independent clusters)
• High α = connections chosen at random
• Small-world networks exist somewhere around the peak (which represents a phase transition)
i.e. where clustering is high but average path length is low
• To the left of the peak, clusters are just starting to join together
• At the peak, everyone is connected
• To the right of the peak, connections are lost as wirings become more random
*Source: Six Degrees, Duncan Watts, 2002, p.78-79 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
40. ▫ Studies conducted by Stanley Milgram beginning
in 1967 at Harvard University
▫ Sent packages to randomly selected people in
Omaha, Nebraska & Wichita
▫ Asked that they bedelivered to individuals in
Milgram repeated other similar
experiments which also received low
Boston, Massachusettscompletion rates
▫ Could only forward package to people they knew
However, experiments on the internet
have since confirmed the number at 6:
on a first-name basis − Facebook application:
▫ Only 64 of 296 letters reached path–=4.5destination
Six Degrees
average
their million users;
5.73
▫ Average path length of these was around 5.5 or 6
▫ Milgram never used the phrase “six degrees of
Some interesting bits: separation” himself
Six degrees of separation
Source: Wikipedia, Small world experiment
Wikipedia, Six degrees of separation An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
41. ▫ The Kevin Bacon game
Aim is to connect all other actors back to Kevin
Bacon
Choice of Kevin Bacon is arbitrary – can be applied
to most actors
Some interesting bits:
Six degrees of separation
*Source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
42. What’s your Erdős number?
(Scientific equivalent of The Kevin Bacon Game)
“Apocalypse” by XKCD
Alt text for this comic:
"I wonder if I still have time to go shoot a short film with
Kevin Bacon?"
URL: http://xkcd.com/599/
Some interesting bits:
Six degrees of separation
43. ▫ A network is considered “scale-free” if its degree
distribution follows a power law
i.e. nodes can have an unlimited number of links to
them e.g. the internet This is what a power law
distribution looks like*
A few nodes have many links, while the majority
have few links
If you take the log of both
axes, you should get a
straight line*
Some interesting bits:
Scale-free networks & power laws
*Image source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
44. ▫ However, very few, if any, networks can display
scale-free properties indefinitely
At some point, limited resources force a cut-off e.g.
limited number of computers in the world
▫ Therefore, generally, scale-free networks only
Taking the log-log of a
display a power law distributionlaw distribution line* area of
power
for some
should show a straight
the graph
However, in practice, the
line is generally only
straight for some area of
the graph*
Some interesting bits:
Scale-free networks & power laws
*Image source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
45. ▫ Power law distributions help us understand
natural growth (e.g. popularity of brands, trends,
ideas, politics, religion, etc.)
Growth in an environment where social influence
occurs tends to result in a power law distribution
(think cumulative advantage)
This comes about due to network behaviour
e.g. nodes with more connections are more likely to
have even more connections (sounds a lot like…
Double Jeopardy!)
▫ This ‘skewing’ of growth patterns is characteristic
Some interesting bits:
Scale-free networks & power laws
of small world networks and results in a few large
components and many small components
An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |
50. •Thanks!
It’s a small world after all
http://www.youtube.com/watch?v=tYQovmtO06k&feature=related
Notas do Editor
New paradigm: “real networks represent populations of individual components that are actually doing something” [Watts, 2002] In other words, networks are dynamic objects that are continually changing Understanding a network is important because its structure affects the individual components’ behaviour and/or the behaviour of the system as a whole Networks are key to understanding non-linear, dynamic systems… … just like those represented by almost every facet of the universe… … from the atomic level right through to the cosmic level The important part is that the components are not acting independently – they are affected by the components around them! Note: links between component may be physical (e.g. power cable, magnetism) or conceptual (e.g. social connections)
Duncan Watts and Steven Strogatz introduced the measure in 1998 Tells you how likely a node is to be connected to its neighbours… … and, importantly, how likely that its neighbours are connected to each other Put another way, it tells you how close a node and its neighbors are to being a clique where “everybody knows everyone else”
Project Description: In 2006, FAS analyzed the director interlock relationships between Fortune 500 companies in California. We looked at how companies are connected through their board of directors, i.e. Apple and Disney are connected through Steve Jobs since he serves on both boards. Companies that share a lot of directors create denser zones in the network and form clusters. We measured which companies exert the largest influence overall and within each cluster. This reveals compelling new insights into key account management. Legend: The triangles represent Fortune 500 companies in CA. The larger the triangle, the more influential the company is. Companies of the same color belong to the same network cluster. If company A and company B share a director, they are linked by a line. The more directors shared, the thicker the line.