(Albert – László Barabási)

1. (Albert – László Barabási)

2. Introduction • Networks are everywhere • Obvious: • Internet • Economy • Social Networks (real and virtual) • Hollywood (6-degrees of Kevin Bacon) • Less Obvious: • The Cell • Genes • Electrical Grids • Formation of Religions

3. Historical Origins of Graph Theory • Invented by Leonhard Euler in 1736 in his mathematical proof of the Königsberg bridge problem:

4. The Random UniverseThe Erdós & Rényi model Avg. 0.5 links per node 15 Nodes 8 Links Avg. 1.0 links per node 15 Nodes 15 Links When nodes are connected randomly and a 50% probability of connectedness is achieved most nodes will be connected When there is an average of one link per node a “super-cluster” is formed Analogous to a phase transition 15 Randomly Distributed Nodes

5. Six Degrees of Separation • Concept first appears in 1929 in the writings of Hungarian poet Frigys Karinthy in a short story from Minden masképpen van (Everything is Different) titled Láncsemek (Chains). **Here estimated at 5 degrees. • Rediscovered in 1967 by Stanley Milgram at MIT with the postcard study. • Postcards sent to random recipients in Wichita, KS and Omaha, NE • Requests that recipients send the package on to someone they deem most likely to know the target persons: • A graduate student in Sharon, MA • A stock broker in Boston, MA • This study finds the average diameter of separation in a human network to be 5.5 (it is estimated to be even smaller today)

6. Diameter of the World Wide Web • By starting on a single webpage and following each of its links, and each link on each of those pages and so on Barabasi and his researchers’ robots begin mapping the web in 1998. • Average of 7 links per page • World Wide Web contains estimated 800-million web pages in 1998 • Diameter of separation on the world wide web is about 18.59 in 1998. (19 clicks)

7. Small Worlds • Strong links in human networks are often not as important as our weak links in connecting us. • The number of links we make is not nearly as important as to whom these links are made. • Andras Schubert coauthorship study proves that the networks we choose to form are, statistically speaking, far from random.

8. Hubs & Connectors • Hollywood is a small world held together by a loosely affiliated web of actors. • Each actor in Hollywood has, on average, 27 links. • However, appearing in a greater number of films does not necessarily ensure an actor’s status as a hub in the Hollywood network. • Six degrees of Kevin Bacon (actually, about 3) • Our networks form hubs of sizes far exceeding statistical prediction. • The Random Network model is flawed.

9. The 80/20 Rule • The perennial 80/20 rule is often an interpretation of power laws. • Power laws are an inherent feature of complex networks

10. The Rich Get Richer • Real networks are governed by two laws: • Growth • Networks grow one node at a time. • Preferential Attachment • Given the choice, new nodes will connect to already well connected nodes. • C:\Documents and Settings\Jeremy Wagner\My Documents\Processing\ScaleFree\ScaleFree_pde\applet\index.html

11. Einstein’s Legacy • Networks can mimic the behavior of rare physical systems such as the Bose-Einstein Condensation predicted by Einstein’s Quantum Theory of Single Atom Gasses • In these systems all atoms are attracted to a single energy state and collapse into a “super-atom” • These systems require advanced laser-cooling to reach the required temperature of 1/1,000,000,000 degree Kelvin. • A single node in the network can take all nodes. • Google • Microsoft Windows • Qwerty

12. Achilles’ Heel • Vulnerability—If our networks are held together by a relative few super-nodes then it ought to be easy to destroy these networks by removing the hubs. • Robustness—In scale-free networks as many as 80% of the nodes can be removed while maintaining links between the remaining 20%. • Cascading failures—common in electrical grids as well as natural systems.

13. Viruses & Fads • The success of a virus or fad is dependent upon the connectedness of its early adopters. • The rapid spread of the AIDS virus is attributable to its propagation within a densely connected web of individuals--some of whom were connected by thousands of links. • The success of a fad is largely a function of its “super-hub” status within a social network.

14. The Awakening Internet • In 1964 Paul Baran began thinking about the optimal structure of the internet and suggests the following topologies: • Centralized –Very Vulnerable • Decentralized --Vulnerable • Distributed –Optimal • The initial design of the internet followed the decentralized model. • Only recently has the internet approached the decentralized model as the private sector has added to it. • The result is an infrastructure that is stretched to its limits beneath the weight of the World Wide Web, email, etc.

15. The Awakening Internet • The internet as an emergent brain. (its already more connected than many animal brains) • Parasitic computing • The Internet as computer? “I can imagine a time when, after getting an answer to a question from your Web browser, neither you nor your computer will know for sure where it came from. After all, do you know where the letter A is stored in your brain?” • The Internet as a living thing • Skin of telemetric sensors • Terabytes of memories • All interconnected

16. The Fragmented Web • Directed Networks complicate (simplify) matters • Continents of a directed network • Directedness allows for targeted self-reinforcement of node sub-networks • Result: rejection of opposing viewpoints and calcification of positions • Ex. Abortion debate websites

17. The Map of Life • Genes form a densely connected web of interactions. • Understanding these connections will lead to new medical therapies and biologic understanding.

18. The Network Economy • There are 10,100 directorships of Fortune 1000 companies held by 7,682 individuals. • 79% serve on one board • 14% serve on two • 7% serve on three or more • Major cluster of 6,724 directors have a diameter of separation of 4.6.(the good old boy network is as real as you think) • Vernon Jordan (former Clinton advisor) is the most connected of these serving on 10 boards, meeting with 106 other directors.

19. The Network Economy • Economic networks’ active reflexivity make them prone to cascading financial failures • Outsourcing requires tight control over supply chain networks. • Traditional business models are incapable of adapting fast enough to respond in these network situations. • Hierarchical thinking does not work in a network economy.

20. Hierarchies & Communities

21. Summary • Networks are integral to the study of complexity. • Understanding complex systems requires an understanding of the underlying network of interactions. • To understand these networks and how they behave we first must understand their topology (we need to map them). • Only after we have such a map can we begin to decipher these complex webs of interaction found in complex networks. • Networks form from simple rule sets which result in complex behaviors. • These aggregate systems adhere to emergent laws that are polymorphisms of other physical systems. • By understanding networks we can better understand our physical world.

22. yn = sin(A*yn-1) – cos(B*xn-1) yn = sin(C*xn-1) – cos(D*yn-1) A = 1.123409 B = 2.412348 C = 1.29 D ~ .987634  2.012879