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

“A collection of objects connected to
each other in some fashion”
[Watts, 2002]

 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


▫ 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

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

Network thinking can be applied
almost anywhere!


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

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

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 |

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


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


Source: The Human Brain Book by Rita Carter

Medicine
e.g. cell formation, nervous system, neural networks


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)


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 |

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

“Unclustered” network “Clustered” network
None of Ego’s friends know each other* All of Ego’s friends know each other


*Source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |

▫ A real-world example: CEOs of Fortune 500 companies
 Which companies share directors? Clusters are
colour-coded


Source: http://flickr.com/photos/11242012@N07/1363558436 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |

▫ 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


Average path length

▫ 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”

Degree distribution

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

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


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 |


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


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


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.


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


End of main
presentation

Next: Interesting
discussion points…

▫ 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

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

 Who spreads ideas?
− Watts vs. Gladwell

vs.
− Mavens/influencers vs. forest fire
− Self-organised criticality
− K-shell decomposition?


How ideas spread

 Which ideas spread?
− Unpredictable
− Ideas that “fit”


How ideas spread

▫ 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


Distributed intelligence

▫ 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!

Distributed intelligence diffusion, memes,
 Works best in small-world networks
 Implications for knowledge

URL: http://www.youtube.com/watch?v=ozkBd2p2piU

Ant colony

Distributed intelligence
Source: http://www.youtube.com/watch?v=ozkBd2p2piU An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |

▫ “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


Random re-wirings
*Source: Six Degrees, Duncan Watts, 2002, p.89 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |


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 |


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”



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


▫ Studies conducted by Stanley Milgram beginning
in 1967 at Harvard University
▫ Sent packages to randomly selected people in
Omaha, Nebraska & Wichita
▫ Asked that they bedelivered 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 |

▫ 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



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/


▫ 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*


Scale-free networks & power laws
*Image source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |

▫ 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*


*Image source: Six Degrees, Duncan Watts, 2002 An Introduction to Network Theory | Kyle Findlay | SAMRA 2010 |

▫ 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

of small world networks and results in a few large
components and many small components

•Software examples…

http://www.youtube.com/watch?v=tYQovmtO06k&feature=related

•Thanks!
It’s a small world after all 

http://www.youtube.com/watch?v=tYQovmtO06k&feature=related

Network theory

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