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The New Science of Networks Lindsay Meyer *Based on the work of Professor Albert- Laszlo Barabasi (Notre Dame)

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Linked Much of my information Comes from this book

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Historical Perspective Konigsberg Bridge Dilemma Connecting 7 bridges Connecting 7 bridges “Can one walk across the 7 bridges and never cross the same path twice?” “Can one walk across the 7 bridges and never cross the same path twice?”

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Eulers Solution: Graph Theory A collection of nodes, connected by links A collection of nodes, connected by links Nodes = pieces of land, Links = bridges Nodes = pieces of land, Links = bridges Nodes with an odd number of links must be the starting or end point of the journey Nodes with an odd number of links must be the starting or end point of the journey Continuous paths may only have 1 starting and 1 end point Continuous paths may only have 1 starting and 1 end point Such a path can NOT exist on a graph that has more than two nodes with an odd number of links Such a path can NOT exist on a graph that has more than two nodes with an odd number of links Konigsburg = 4 nodes, no path Konigsburg = 4 nodes, no path

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Eulers Take-Home Message “Graphs or networks have hidden properties in their construction that limit or enhance our ability to do things with them” “Graphs or networks have hidden properties in their construction that limit or enhance our ability to do things with them” A sudden change in layout can help remove constraints A sudden change in layout can help remove constraints IE: Building a new bridge and increasing the the number of links of two nodes to four (an even number) IE: Building a new bridge and increasing the the number of links of two nodes to four (an even number)

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Social Networks: The Party… The expensive, unlabeled wine scenario The expensive, unlabeled wine scenario 100 guests which cluster into groups of 2- 3 people 100 guests which cluster into groups of 2- 3 people These people mingle… These people mingle…

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So you have “mingling”… Suddenly, people begin moving on to other social clusters, but there are invisible links between those who initiated contact with each other Suddenly, people begin moving on to other social clusters, but there are invisible links between those who initiated contact with each other Subtle paths connect people to each other… the “secret” gets out as people share this special knowledge with their new friends Subtle paths connect people to each other… the “secret” gets out as people share this special knowledge with their new friends Erdos & Renyi: 30 mins and everyone in the room is somehow connected. “If each person gets to know one other guest, then soon everyone will be drinking the reserve port!” Erdos & Renyi: 30 mins and everyone in the room is somehow connected. “If each person gets to know one other guest, then soon everyone will be drinking the reserve port!”

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Other examples of networks Remember, a network is a bunch of nodes connected by links Remember, a network is a bunch of nodes connected by links Computers – Phone lines Computers – Phone lines Molecules – Biochemical rxns Molecules – Biochemical rxns Companies – Consumers (trade) Companies – Consumers (trade) Nerve cells – Axons Nerve cells – Axons Islands – Bridges Islands – Bridges

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That MAGIC MOMENT!!! “The moment when your expensive wine is in DANGER” “The moment when your expensive wine is in DANGER” Mathematicians call it the emergence of a giant component Mathematicians call it the emergence of a giant component Physicists call it percolation and explain it with phase change Physicists call it percolation and explain it with phase change Sociologists would say that a community formed Sociologists would say that a community formed The big picture: when we randomly pick and connect nodes together, something special happens. Before it’s a bunch of tiny isolated clusters and after, nearly everyone is joined!

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6 Degrees of Separation Milgrams experiment to see how connected people were between distant cities (ie: Omaha to Boston) Milgrams experiment to see how connected people were between distant cities (ie: Omaha to Boston) HOW HE DID IT: 1. Sent out letters with postcards to be returned to Harvard 2. Stipulation: If you did not know the target, then forward the letter on to someone who might have better odds of knowing the person THAT YOU KNOW 3. If you know the target, mail the folder directly to the person *The results? One letter only took two steps, but on average, it took 5.5 people to make it to the target person (with 42 of the original 160 letters actually returning to Cambridge)

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1 2 3 4 5 6 So perhaps this isn’t actually accurate, for we are all connected… by the means of our class (ps. Thank you www.thefacebook.com)www.thefacebook.com 6 Degrees of Separation?!

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Scale-Free Networks & 80-20 “Various complex systems have an underling architecture governed by shared organizing principles” “Various complex systems have an underling architecture governed by 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 to other nodes (and are known as hubs) and these networks are scale-free Some nodes have tons of connections to other nodes (and are known as hubs) and these networks are scale-free Characteristics include: highly robust yet very vulnerable to coordinated attack Characteristics include: highly robust yet very vulnerable to coordinated attack

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Examples, please… SCALE FREE NETWORKS

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So about this 80-20 thing? 80% of peas are produced by 20% of peapods 80% of peas are produced by 20% of peapods 80% of the land in Italy is owned by 20% of the population 80% of the land in Italy is owned by 20% of the population 80% of profits are produced by 20% of the employees (Murphy’s Law of Management) 80% of profits are produced by 20% of the employees (Murphy’s Law of Management) 80% of customer service problems are created by 20% of consumers 80% of customer service problems are created by 20% of consumers 80% of decisions are made during 20% of meeting time 80% of decisions are made during 20% of meeting time 80% of crime is committed by 20% of individuals 80% of crime is committed by 20% of individuals

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

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Versus Power Law In networks the power law describes the degree distribution In networks the power law describes the degree distribution The exponent is the degree exponent; there is no peak The exponent is the degree exponent; there is no peak Consider the internet and links from webpage to webpage, an obvious network Consider the internet and links from webpage to webpage, an obvious network Number of web pages with k incoming links: N(k) ~ k -γ Number of web pages with k incoming links: N(k) ~ k -γ Slope of line of log-log plot = 2.1 Slope of line of log-log plot = 2.1 Outgoing = 2.5 Outgoing = 2.5

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Network Governance Two laws: growth and preferential attachment Two laws: growth and preferential attachment Assume that new nodes connect via two links (will always choose the node with more connections) Assume that new nodes connect via two links (will always choose the node with more connections) This is how we get the “highly connected hubs” and the power law is modeled This is how we get the “highly connected hubs” and the power law is modeled “Rich-get-richer” phenomenon “Rich-get-richer” phenomenon

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

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The Barabási-Albert [BA] model (1999) ER Model Look at the distribution of degrees ER ModelWS Model actorspower grid www The probability of finding a highly.

The Barabási-Albert [BA] model (1999) ER Model Look at the distribution of degrees ER ModelWS Model actorspower grid www The probability of finding a highly.

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