An Introduction to Network Theory
Kyle Findlay [email_address] 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

What is a network? <ul><ul><li>An
agent/object's actions are affected by the actions of others around it. </li></ul></ul><ul><ul><li>Actions, choices, etc. are not made in isolation i.e. they are contingent on others' actions, choices, etc. </li></ul></ul>

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

<ul><ul><li>Social groups </li></ul></ul><ul><ul><li>The internet </li></ul></ul><ul><ul><li>Cities
</li></ul></ul>What is a network? <ul><ul><li>Diseases </li></ul></ul><ul><ul><li>Neural networks (computer & human) </li></ul></ul><ul><ul><li>Proteins & genes </li></ul></ul><ul><ul><li>Stem cells </li></ul></ul><ul><ul><li>Other cells </li></ul></ul><ul><ul><li>Plants </li></ul></ul>Source: Six Degrees, Duncan Watts, 2002 The blogosphere Proliferation of landlines in London Human genome Quaking Aspen (one of the largest organisms in the world – these trees represent a single organism with a shared root system) Rabbit cell

<ul><ul><li>New paradigm : “real networks
represent populations of individual components that are actually doing something ” [Watts, 2002] </li></ul></ul><ul><ul><ul><li>In other words, networks are dynamic objects that are continually changing </li></ul></ul></ul><ul><ul><ul><li>Understanding a network is important because its structure affects the individual components’ behaviour and the behaviour of the system as a whole </li></ul></ul></ul><ul><ul><li>Networks are key to understanding non-linear , dynamic systems … </li></ul></ul><ul><ul><ul><li>… just like those represented by almost every facet of the universe … </li></ul></ul></ul><ul><ul><ul><li>… from the atomic level right through to the cosmic level </li></ul></ul></ul><ul><ul><li>The important part is that the components are not acting independently – they are affected by the components around them ! </li></ul></ul><ul><ul><li>Note: links between component may be physical (e.g. power cable, magnetism) or conceptual (e.g. social connections) </li></ul></ul>Networks used to be thought of as fixed structures What is a network?

Shipping (sea) networks Telecommunications networks
Air traffic network Data networks Source: Britain From Above ( http:// www.bbc.co.uk/britainfromabove ) CAUTION: Gratuitous network shots

Network thinking can be applied
almost anywhere !

<ul><li>Epidemiology (i.e. spread of diseases)
</li></ul><ul><ul><li>e.g. spread of foot & mouth disease in the UK in 2001 over 75 days </li></ul></ul>Where’s it applied? URL: http:// www.youtube.com/watch?v =PufTeIBNRJ4

<ul><li>Physics </li></ul><ul><ul><li>e.g. particle interactions, the
structure of the universe </li></ul></ul>Where’s it applied? URL: http:// www.youtube.com/watch?v =8C_dnP2fvxk

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? URL: http:// www.youtube.com/watch?v =lRZ2iEHFgGo URL: http:// www.youtube.com/watch?v = AEoP-XtJueo

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? URL: http:// www.youtube.com/watch?v =l-RoDv7c5ok URL: http:// www.youtube.com/watch?v =o4g930pm8Ms Vid not working

Biology e.g. fish swimming in
schools, ant colonies, birds flying in formation, crickets chirping in unison, giant honeybees shimmering Where’s it applied? URL: http:// www.youtube.com/watch?v =Sp8tLPDMUyg URL: http:// www.youtube.com/watch?v =YadP3w7vkJA

Medicine e.g. cell formation, nervous
system, neural networks Where’s it applied? Source: The Human Brain Book by Rita Carter

And, most interestingly…society e.g. interactions
between people (e.g. Facebook; group behaviour) Where’s it applied? URL: http:// www.youtube.com/watch?v =9n9irapdON4 URL: http:// www.youtube.com/watch?v =sD2yosZ9qDw

<ul><ul><li>Node = individual components of
a network e.g. people, power stations, neurons, etc. </li></ul></ul><ul><ul><li>Edge = direct link between components (referred to as a dyad in context of social networking i.e. relationship between two people) </li></ul></ul><ul><ul><li>Path = route taken across components to connect two nodes </li></ul></ul>Some terminology…

<ul><ul><li>Tells you how likely a
node is to be connected to its neighbours … </li></ul></ul><ul><ul><ul><li>… and, importantly, how likely that its neighbours are connected to each other </li></ul></ul></ul><ul><ul><li>Put another way, it tells you how close a node and its neighbors are to being a clique where “everybody knows everyone else” </li></ul></ul>Important network features: Clustering co-efficient <ul><ul><li>No connections </li></ul></ul><ul><ul><li>between nodes </li></ul></ul>c = 0 Some nodes connected c = 1/3 <ul><ul><li>All relevant nodes connected </li></ul></ul>c = 1

<ul><ul><li>“ Unclustered” network </li></ul></ul><ul><ul><li>“ Clustered”
network </li></ul></ul>*Source: Six Degrees, Duncan Watts, 2002 None of Ego’s friends know each other* All of Ego’s friends know each other Important network features: Clustering co-efficient

<ul><ul><li>A real-world example: CEOs of
Fortune 500 companies </li></ul></ul><ul><ul><ul><li>Which companies share directors? Clusters are colour-coded </li></ul></ul></ul>Source: http://flickr.com/photos/11242012@N07/1363558436 Important network features: Clustering co-efficient

<ul><ul><li>Average path length = the
average number of ‘hops’ required to reach any other node in the network </li></ul></ul><ul><ul><li>“ Six degrees of separation ” average path length = 6 </li></ul></ul>Important network features: Average path length

<ul><ul><li>The degree of a node
is the number of connections (or edges) it has coming in from, and going out to, other nodes </li></ul></ul>Node 10 connections or “ edges ” Important network features: Degree distribution 3 2 1 4 5 6 7 8 10 9

<ul><ul><li>They sit on a continuum
based on a few factors: </li></ul></ul>*Source: Six Degrees, Duncan Watts, 2002 3 main types of networks 2 Clustering 3 Ave path length 1 Randomness <ul><ul><li>Grid/lattice network (structure, order) </li></ul></ul><ul><ul><li>Small-world network (a mix of order and randomness) </li></ul></ul><ul><ul><li>Random network (randomness) </li></ul></ul><< Level of randomness of links >> β=0 β=1

<ul><ul><li>Simplest form of network with
nodes ranged geometrically </li></ul></ul><ul><ul><li>Low degree (nodes only connected to closest neighbours ) </li></ul></ul><ul><ul><li>High clustering </li></ul></ul><ul><ul><li>Long average path length (no shortcuts – have to go through all nodes) </li></ul></ul><ul><ul><li>Pros : methodical , easy to visualise </li></ul></ul><ul><ul><li>Cons : not very good at modeling most real-world networks </li></ul></ul>3 main types of networks: Grid/Lattice network Molecule 1D lattice Diamond (crystal) lattice Bismuth crystal

<ul><ul><li>Most nodes aren’t neighbours ,
but they can be reached from every other node by a small number of hops or steps </li></ul></ul><ul><ul><ul><li>i.e. small average path length due to a few random re-wirings </li></ul></ul></ul>3 main types of networks: Small world network Source: http://en.wikipedia.org/wiki/Small-world_network <ul><ul><li>Higher clustering co-efficient than one would expect if connections were made by pure random chance </li></ul></ul><ul><ul><ul><li>“ 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 ” </li></ul></ul></ul><ul><ul><li>Common in nature , including everything from the internet to gene regulatory networks to ecosystems </li></ul></ul>

<ul><ul><li>Lower clustering than small-world networks
generally </li></ul></ul><ul><ul><li>No “force” or “bias” influencing how links are created between nodes </li></ul></ul><ul><ul><ul><ul><li>i.e. probability of creating an edge/link is independent of previous connections </li></ul></ul></ul></ul>3 main types of networks: Random network

<ul><ul><li>Nature </li></ul></ul><ul><ul><li>Networks are evident everywhere
in nature </li></ul></ul><ul><ul><li>In fact, most natural growth processes come about due to network behaviour </li></ul></ul>The big picture <ul><ul><li>Natural growth = evolutionary , iterative growth , where future growth is constrained by previous growth patterns (referred to as path dependence ) </li></ul></ul><ul><ul><ul><li>i.e. growth follows the network structure </li></ul></ul></ul>

<ul><ul><li>Networks better reflect reality and
capture complexity i.e. non-linear dynamics </li></ul></ul><ul><ul><li>Network theory helps us to better understand: </li></ul></ul>And market research? How will word of my brand permeate through my target market? ? How will negative publicity about my brand spread and be interpreted? ? Who are the gatekeepers in a community that most affect the flow of information ? ? <ul><ul><ul><li>How is the market likely to fall out in terms of </li></ul></ul></ul><ul><ul><ul><li>market share (Double Jeopardy)? </li></ul></ul></ul>? <ul><ul><ul><li>What will the non-linear impact be of a specific change in the market </li></ul></ul></ul><ul><ul><ul><li>e.g. change in market share, perceptions, etc. </li></ul></ul></ul>?

<ul><ul><li>Network theory has been used
to understand imagery and market barriers </li></ul></ul><ul><ul><ul><li>Adjusting attributes and seeing knock-on effect in network </li></ul></ul></ul><ul><ul><ul><li>Using agent-based modeling to model this effect </li></ul></ul></ul>And market research? <ul><ul><li>Useful for word-of-mouth / viral approaches </li></ul></ul><ul><ul><ul><li>Watts and Peretti use network theory to increase reach of WOM campaigns </li></ul></ul></ul><ul><ul><li>Helps us avoid thinking about things in a vacuum as it takes account of inter-related variables… </li></ul></ul><ul><ul><ul><li>… and provides us with counter-intuitive outcomes that we may not have reached on our own </li></ul></ul></ul>

Next: Interesting discussion points… End
of main presentation

<ul><ul><li>This is a very simple
2D representation of how I roughly visualise information propagating through a network </li></ul></ul><ul><ul><ul><li>It is very simple and doesn’t take into account many concepts </li></ul></ul></ul><ul><ul><ul><li>But it is a visual aid that helps one to start thinking about how information might spread from person to person </li></ul></ul></ul>Some interesting bits: A network in action Source: http://www.funny-games.biz/reaction-effect.html

<ul><ul><li>What does a highly “spreadable”
idea look like? </li></ul></ul><ul><ul><li>1,636,967 views in two months (as at 25 April 2010) </li></ul></ul><ul><ul><li>Performed in front of 75,000 people at Coachella Music Festival (California, April 2010) </li></ul></ul>URL: http:// www.youtube.com/watch?v =Q77YBmtd2Rw Some interesting bits: How ideas spread

Some interesting bits: How ideas
spread <ul><ul><li>Who spreads ideas? </li></ul></ul><ul><ul><ul><li>Watts vs. Gladwell </li></ul></ul></ul><ul><ul><ul><li>Mavens/influencers vs. forest fire </li></ul></ul></ul><ul><ul><ul><li>Self-organised criticality </li></ul></ul></ul><ul><ul><ul><li>K-shell decomposition? </li></ul></ul></ul>vs.

<ul><ul><li>Which ideas spread? </li></ul></ul><ul><ul><ul><li>Unpredictable </li></ul></ul></ul><ul><ul><ul><li>Ideas
that “fit” </li></ul></ul></ul>Some interesting bits: How ideas spread

<ul><ul><li>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 only focused on self-interest without the bigger picture in mind </li></ul></ul>Some interesting bits: Distributed intelligence <ul><ul><ul><li>e.g. no single ant knows how to build an ant colony </li></ul></ul></ul><ul><ul><ul><li>e.g. the internet has grown organically over time with no single person directing its growth </li></ul></ul></ul><ul><ul><ul><li>i.e. no grand designer </li></ul></ul></ul><ul><ul><li>This is known as self-organisation and/or emergence , and is a property of complex networks and non-linear, dynamic systems </li></ul></ul>

<ul><ul><li>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! </li></ul></ul><ul><ul><ul><li>Works best in small-world networks </li></ul></ul></ul><ul><ul><ul><li>Implications for knowledge diffusion , memes , collaboration , etc. </li></ul></ul></ul>Some interesting bits: Distributed intelligence

Some interesting bits: Distributed intelligence
Ant colony Source: http:// www.youtube.com/watch?v =ozkBd2p2piU URL: http:// www.youtube.com/watch?v =ozkBd2p2piU

<ul><ul><li>“ 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]* </li></ul></ul>*Source: Six Degrees, Duncan Watts, 2002, p.89 Random re-wirings Long average path length Dramatically reduced average path length Some interesting bits: Random re-wirings “ 8” “ 3”

<ul><ul><li>People are more likely to
become acquainted over time when they have something in common </li></ul></ul><ul><ul><ul><li>i.e. we have a bias towards the familiar , thus reducing the pure randomness of connections </li></ul></ul></ul><ul><ul><ul><li>Known as “ homophily ” - “birds of a feather flock together” </li></ul></ul></ul><ul><ul><li>Network connections don’t arise independently of each other… </li></ul></ul><ul><ul><ul><li>… they are influenced by previous connections </li></ul></ul></ul><ul><ul><li>If A knows B… </li></ul></ul><ul><ul><ul><li>… and B knows C… </li></ul></ul></ul><ul><ul><ul><ul><li>… then A is much more likely to know C </li></ul></ul></ul></ul>*Source: Six Degrees, Duncan Watts, 2002, p.58-61 A B C Some interesting bits: Triadic closure

*Source: Six Degrees, Duncan Watts,
2002, p.58-61 A B C <ul><ul><li>This is why random re-wirings are so effective at reducing the ave. path length … </li></ul></ul><ul><ul><ul><li>… they help connect clusters, or ‘cliques’, that might otherwise exist in isolation </li></ul></ul></ul><ul><ul><li>This is the strength of the small-world network: </li></ul></ul><ul><ul><ul><li>High clustering and a relatively small amount of random re-wirings allows for a dramatically reduced average path length… </li></ul></ul></ul><ul><ul><ul><li>… allowing everyone to connect to everyone else in relatively few steps e.g. “six degrees of separation” </li></ul></ul></ul>Some interesting bits: Triadic closure

*Source: Six Degrees, Duncan Watts,
2002, p.78-79 <ul><ul><li>This “ birds of a feather flock together ” effect was modeled by Watts and Strogatz* </li></ul></ul><ul><ul><ul><li>They used α (alpha) to represent level of preference to only connect with friends of friends </li></ul></ul></ul><ul><ul><ul><li>Low α = strong preference to only connect with friends of friends (triadic closures occur, independent clusters) </li></ul></ul></ul><ul><ul><ul><li>High α = connections chosen at random </li></ul></ul></ul><ul><li>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 </li></ul><ul><ul><ul><li>To the left of the peak, clusters are just starting to join together </li></ul></ul></ul><ul><ul><ul><li>At the peak , everyone is connected </li></ul></ul></ul><ul><ul><ul><li>To the right of the peak, connections are lost as wirings become more random </li></ul></ul></ul>Some interesting bits: Triadic closure

<ul><ul><li>Studies conducted by Stanley Milgram
beginning in 1967 at Harvard University </li></ul></ul><ul><ul><li>Sent packages to randomly selected people in Omaha , Nebraska & Wichita </li></ul></ul><ul><ul><li>Asked that they be delivered to individuals in Boston , Massachusetts </li></ul></ul><ul><ul><li>Could only forward package to people they knew on a first-name basis </li></ul></ul><ul><ul><li>Only 64 of 296 letters reached their destination </li></ul></ul><ul><ul><li>Average path length of these was around 5.5 or 6 </li></ul></ul><ul><ul><li>Milgram never used the phrase “six degrees of separation” himself </li></ul></ul>Source: Wikipedia, Small world experiment Wikipedia, Six degrees of separation <ul><ul><li>Milgram repeated other similar experiments which also received low completion rates </li></ul></ul><ul><ul><li>However, experiments on the internet have since confirmed the number at 6: </li></ul></ul><ul><ul><ul><li>Facebook application: Six Degrees – 4.5 million users ; average path = 5.73 </li></ul></ul></ul>Some interesting bits: Six degrees of separation

<ul><ul><li>The Kevin Bacon game </li></ul></ul><ul><ul><ul><li>Aim
is to connect all other actors back to Kevin Bacon </li></ul></ul></ul><ul><ul><ul><li>Choice of Kevin Bacon is arbitrary – can be applied to most actors </li></ul></ul></ul>*Source: Six Degrees, Duncan Watts, 2002 Some interesting bits: Six degrees of separation

Some interesting bits: Six degrees
of separation What’s your Erdős number? (Scientific equivalent of The Kevin Bacon Game) “ Apocalypse” by XKCD Alt text for this comic: &quot;I wonder if I still have time to go shoot a short film with Kevin Bacon?&quot; URL: http://xkcd.com/599/

<ul><ul><li>A network is considered “
scale-free ” if its degree distribution follows a power law </li></ul></ul><ul><ul><ul><li>i.e. nodes can have an unlimited number of links to them e.g. the internet </li></ul></ul></ul><ul><ul><ul><li>A few nodes have many links , while the majority have few links </li></ul></ul></ul>This is what a power law distribution looks like* If you take the log of both axes, you should get a straight line* *Image source: Six Degrees, Duncan Watts, 2002 Some interesting bits: Scale-free networks & power laws

<ul><ul><li>However, very few , if
any, networks can display scale-free properties indefinitely </li></ul></ul><ul><ul><ul><li>At some point, limited resources force a cut-off e.g. limited number of computers in the world </li></ul></ul></ul><ul><ul><li>Therefore, generally, scale-free networks only display a power law distribution for some area of the graph </li></ul></ul>Taking the log-log of a power law distribution should show a straight line* However, in practice, the line is generally only straight for some area of the graph* *Image source: Six Degrees, Duncan Watts, 2002 Some interesting bits: Scale-free networks & power laws

<ul><ul><li>Power law distributions help us
understand natural growth (e.g. popularity of brands, trends, ideas, politics, religion, etc.) </li></ul></ul><ul><ul><ul><li>Growth in an environment where social influence occurs tends to result in a power law distribution (think cumulative advantage ) </li></ul></ul></ul><ul><ul><ul><li>This comes about due to network behaviour </li></ul></ul></ul><ul><ul><ul><ul><li>e.g. nodes with more connections are more likely to have even more connections (sounds a lot like… Double Jeopardy !) </li></ul></ul></ul></ul><ul><ul><li>This ‘ skewing ’ of growth patterns is characteristic of small world networks and results in a few large components and many small components </li></ul></ul><ul><ul><li>In order to understand non-linear growth/dynamics, we need to understand the influence of the network on the individual components </li></ul></ul><ul><ul><li>Power laws help us describe this influence </li></ul></ul><ul><ul><ul><ul><li>And networks help us understand how power laws come about </li></ul></ul></ul></ul>Some interesting bits: Scale-free networks & power laws

<ul><li>Software examples… </li></ul><ul><ul><li>http://www.youtube.com/watch?v=tYQovmtO06k&feature=related </li></ul></ul>

<ul><li>Thanks! </li></ul><ul><ul><li>http://www.youtube.com/watch?v=tYQovmtO06k&feature=related </li></ul></ul>It’s a small
world after all 