The New Science of The New Science of NetworksNetworks

Lindsay MeyerLindsay Meyer*Based on the work of Professor Albert-*Based on the work of Professor Albert-

Laszlo Barabasi (Notre Dame)Laszlo Barabasi (Notre Dame)

LinkedLinked

Much of my informationComes from this book

Historical PerspectiveHistorical Perspective

Konigsberg Bridge DilemmaKonigsberg Bridge Dilemma Connecting 7 bridgesConnecting 7 bridges ““Can one walk across the 7 bridges and Can one walk across the 7 bridges and

never cross the same path twice?”never cross the same path twice?”

Eulers Solution: Graph TheoryEulers Solution: Graph Theory

A collection of nodes, connected by linksA collection of nodes, connected by links Nodes = pieces of land, Links = bridgesNodes = pieces of land, Links = bridges Nodes with an odd number of links must be Nodes with an odd number of links must be

the starting or end point of the journeythe starting or end point of the journey Continuous paths may only have 1 starting Continuous paths may only have 1 starting

and 1 end pointand 1 end point Such a path can NOT exist on a graph that Such a path can NOT exist on a graph that

has more than two nodes with an odd has more than two nodes with an odd number of linksnumber of links

Konigsburg = 4 nodes, no pathKonigsburg = 4 nodes, no path

Eulers Take-Home MessageEulers Take-Home Message

““Graphs or networks have hidden Graphs or networks have hidden properties in their construction that properties in their construction that limit or enhance our ability to do limit or enhance our ability to do things with them”things with them”

A sudden change in layout can help A sudden change in layout can help remove constraintsremove constraints

IE: Building a new bridge and IE: Building a new bridge and increasing the the number of links of increasing the the number of links of two nodes to four (an even number)two nodes to four (an even number)

Social Networks: The Party…Social Networks: The Party…

The expensive, The expensive, unlabeled wine unlabeled wine scenarioscenario

100 guests 100 guests which cluster which cluster into groups of 2-into groups of 2-3 people3 people

These people These people mingle…mingle…

So you have “mingling”…So you have “mingling”…

Suddenly, people begin moving on to other Suddenly, people begin moving on to other social clusters, but there are invisible links social clusters, but there are invisible links between those who initiated contact with between those who initiated contact with each othereach other

Subtle paths connect people to each Subtle paths connect people to each other… the “secret” gets out as people other… the “secret” gets out as people share this special knowledge with their new share this special knowledge with their new friendsfriends

Erdos & Renyi: 30 mins and everyone in Erdos & Renyi: 30 mins and everyone in the room is somehow connected. “If each the room is somehow connected. “If each person gets to know one other guest, then person gets to know one other guest, then soon everyone will be drinking the reserve soon everyone will be drinking the reserve port!”port!”

Other examples of networksOther examples of networks

Remember, a network is a Remember, a network is a bunch of nodes connected by bunch of nodes connected by linkslinks Computers – Phone linesComputers – Phone lines Molecules – Biochemical rxnsMolecules – Biochemical rxns Companies – Consumers (trade)Companies – Consumers (trade) Nerve cells – AxonsNerve cells – Axons Islands – BridgesIslands – Bridges

That That !!!!!!

““The moment when your expensive The moment when your expensive wine is in DANGER”wine is in DANGER” Mathematicians call it the emergence of Mathematicians call it the emergence of

a giant componenta giant component Physicists call it percolation and explain Physicists call it percolation and explain

it with phase changeit with phase change Sociologists would say that a community Sociologists would say that a community

formedformedThe big picture: when we randomly pick The big picture: when we randomly pick

and connect nodes together, something and connect nodes together, something special happens. Before it’s a bunch of special happens. Before it’s a bunch of tiny isolated clusters and after, nearly tiny isolated clusters and after, nearly everyone is joined!everyone is joined!

6 Degrees of Separation6 Degrees of Separation

MilgramsMilgrams experiment to see how connected experiment to see how connected people were between distant cities (ie: Omaha to people were between distant cities (ie: Omaha to Boston)Boston)

HOW HE DID IT:HOW HE DID IT:1.1. Sent out letters with postcards to be returned to Sent out letters with postcards to be returned to

HarvardHarvard2.2. Stipulation: If you did not know the target, then Stipulation: If you did not know the target, then

forward the letter on to someone who might have forward the letter on to someone who might have better odds of knowing the person THAT YOU better odds of knowing the person THAT YOU KNOWKNOW

3.3. If you know the target, mail the folder directly to If you know the target, mail the folder directly to the personthe person

*The results? One letter only took two steps, but on *The results? One letter only took two steps, but on average, it took 5.5 people to make it to the target average, it took 5.5 people to make it to the target person (with 42 of the original 160 letters actually person (with 42 of the original 160 letters actually returning to Cambridge)returning to Cambridge)

1

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5

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So perhaps this isn’t actually accurate, for we are all connected… by

the means of our class (ps. Thank you www.thefacebook.com)

6 Degrees of Separation?!

Scale-Free Networks & 80-20Scale-Free Networks & 80-20

““Various complex systems have an Various complex systems have an underling architecture governed by underling architecture governed by shared organizing principles”shared organizing principles”

We know this stuff like the pro’s:We know this stuff like the pro’s: Some nodes have tons of connections Some nodes have tons of connections

to other nodes (and are known as hubs) to other nodes (and are known as hubs) and these networks are scale-freeand these networks are scale-free

Characteristics include: highly robust yet Characteristics include: highly robust yet very vulnerable to coordinated attackvery vulnerable to coordinated attack

Examples, please…Examples, please…

SCALE FREE NETWORKS

So about this 80-20 thing?So about this 80-20 thing?

80% of peas are produced by 20% of 80% of peas are produced by 20% of peapodspeapods

80% of the land in Italy is owned by 20% of 80% of the land in Italy is owned by 20% of the populationthe population

80% of profits are produced by 20% of the 80% of profits are produced by 20% of the employees (Murphy’s Law of Management)employees (Murphy’s Law of Management)

80% of customer service problems are 80% of customer service problems are created by 20% of consumerscreated by 20% of consumers

80% of decisions are made during 20% of 80% of decisions are made during 20% of meeting timemeeting time

80% of crime is committed by 20% of 80% of crime is committed by 20% of individualsindividuals

Bell CurveBell Curve

Many things in nature are follow Many things in nature are follow a “normal distribution” or bell a “normal distribution” or bell curve with empirical rule: curve with empirical rule: http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.hthttp://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.htmlml

Versus Power LawVersus Power Law

In networks the power law describes In networks the power law describes the degree distributionthe degree distribution

The exponent is the degree The exponent is the degree exponent; there is no peakexponent; there is no peak

Consider the internet and links from Consider the internet and links from webpage to webpage, an obvious webpage to webpage, an obvious networknetwork Number of web pages with k incoming Number of web pages with k incoming

links: N(k) ~ klinks: N(k) ~ k-γ-γ

Slope of line of log-log plot = 2.1Slope of line of log-log plot = 2.1 Outgoing = 2.5 Outgoing = 2.5

Network GovernanceNetwork Governance

Two laws: Two laws: growthgrowth and and

Assume that new nodes connect via Assume that new nodes connect via

two links (will always choose the node two links (will always choose the node with more connections)with more connections)

This is how we get the “highly This is how we get the “highly connected hubs” and the power law is connected hubs” and the power law is modeledmodeled

““Rich-get-richer”Rich-get-richer” phenomenon phenomenon

From From NetworksNetworks……

““The goal before us is to understand complexity.The goal before us is to understand complexity. To achieve that, we move beyond structure and To achieve that, we move beyond structure and topology and start focusing on the dynamics that topology and start focusing on the dynamics that take place along the links. take place along the links. Networks are only the Networks are only the skeleton of complexity, the highways for the skeleton of complexity, the highways for the various processes that make our world humvarious processes that make our world hum… Our … Our quest to understand nature has hit a glass ceiling quest to understand nature has hit a glass ceiling because because we do not yet know how to fit the pieces we do not yet know how to fit the pieces togethertogether. The complex issues with which we are . The complex issues with which we are faced, in fields from communications systems to faced, in fields from communications systems to cell biology, demand a brand new framework… cell biology, demand a brand new framework… Now we must follow these maps to complete the Now we must follow these maps to complete the journey, fitting the pieces to one another, node by journey, fitting the pieces to one another, node by node and link by link, and capturing their dynamic node and link by link, and capturing their dynamic interplayinterplay.”.”~Albert – Laszla Barabasi, “Linked” pp. 225-226~Albert – Laszla Barabasi, “Linked” pp. 225-226

To To ComplexityComplexity!!!!!!