The New Science of Networks

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

2. Linked Much of my information Comes from this book

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

4. Eulers Solution: Graph Theory • A collection of nodes, connected by links • Nodes = pieces of land, Links = bridges • 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 • 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

5. Eulers Take-Home Message • “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 • IE: Building a new bridge and increasing the the number of links of two nodes to four (an even number)

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

7. 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 • 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!”

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

9. That MAGIC MOMENT!!! • “The moment when your expensive wine is in DANGER” • Mathematicians call it the emergence of a giant component • Physicists call it percolation and explain it with phase change • 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!

10. 6 Degrees of Separation • Milgrams experiment to see how connected people were between distant cities (ie: Omaha to Boston) HOW HE DID IT: • Sent out letters with postcards to be returned to Harvard • 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 • 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)

11. 6 Degrees of Separation?! 2 5 4 1 6 3 So perhaps this isn’t actually accurate, for we are all connected… by the means of our class  (ps. Thank you www.thefacebook.com)

12. Scale-Free Networks & 80-20 • “Various complex systems have an underling architecture governed by shared organizing principles” • 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 • Characteristics include: highly robust yet very vulnerable to coordinated attack

13. Examples, please… SCALE FREE NETWORKS

14. So about this 80-20 thing? • 80% of peas are produced by 20% of peapods • 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 customer service problems are created by 20% of consumers • 80% of decisions are made during 20% of meeting time • 80% of crime is committed by 20% of individuals

15. 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

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

17. Network Governance • Two laws: growthand preferential attachment • 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 • “Rich-get-richer” phenomenon

18. 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.” ~Albert – Laszla Barabasi, “Linked” pp. 225-226 To Complexity!!!