By: Eric Havens, Sanusha Matthews, and Mike Copciac R ICH GET RICHER

O LD N ETWORK A SSUMPTIONS  1) Having all the nodes from the beginning, we assume that the number of nodes is fixed and remains unchanged throughout the network’s life.  2) All nodes are equivalent  For nearly fourty years of network research these assumptions were unquestioned. The discovery of hubs and the power laws that describe them, forced us to abandon both assumptions.

G ROWTH : THE FEATURE MOST NETWORKS HAVE IN COMMON  If you look at any network you will likely see that starting with a few nodes, it grew incrementally through the addition of new nodes, gradually reaching its current size.  World Wide Web-started with only one node (website) which was by Tim Berners-Lee. Physicists and computer scientists started creating pages of their own and within 10 years there were thousands of websites.  Hollywood network- had only 53 actors in 1900 and has grown to over half a million.

R ANDOM N ETWORK VS. P REFERENTIAL A TTACHMENT  Random Network  Choosing a news site off the internet- Yahoo’s directory offers over 8,000 news sources and each are equally likely to be chosen based on this theory.  Picking an actor for a role in a movie-each of the thousands of actors has an equal change of being chosen  Preferential Attachment  We choose big news outlets or the ones we are most familiar with.  A director chooses based on how well they fit the role and popularity. The ones that have been in the most movies are the most likely to be selected (rich get richer)

T HE B IRTH OF A S CALE - F REE N ETWORK  From the two key concepts of growth and preferential attachment in networks the scale- free topology is a natural consequence of the continuously expanding nature of real networks.  When deciding where to link, new nodes prefer to attach to the more connected nodes. Due to growth and preferential attachment, a few highly connected hubs emerge.

 The topology of real network was shaped by many effect like  All links present in the scale free model are added when new nodes join the network, in most network new links can emerge spontaneously.  In many network nodes and links can disappear. Indeed,many web pages go out of business, taking with them thousands of links.  Links can also be rewired.

 Luis Amaral, a research professor at Boston university demonstrated that  If nodes fail to acquire links after a certain age the size of the hubs will be eliminated, making large hubs less frequent than predicted by a power law.  Assuming that nodes slowly lose their ability to attract as they age Mendes and Dorogovtsev showed that gradual aging does not destroy power laws, but merely alters the number of hubs by changing the degree exponent.

 Paul Krapivsky and Sid Redner from Boston university found that linking to a node would not be simply proportional to the number of links the node has but would follow some more complicated function. They also found that such effect can destroy the power law characterizing the network.

T HE EIGHTH LEGACY Einstein’s legacy

 Google launched in 1997, was a latecomer to the web.  It violated the basic prediction of the scale-free model, that the first mover has an advantage.  It became the both the biggest node and the most popular search engine.

 In a competitive environment each node has a certain fitness  Fitness is a quantitative measure of a node’s ability to stay in front of the competition  Nodes with higher fitness are linked more frequently.  Between two nodes with the same number of links, the fitter one acquires links more quickly.  If two nodes have the same fitness, the older one has an advantage

 Independent of when a node joins the network, a fit node will soon leave behind all nodes with smaller fitness  e.g. Google a late comer with great searching technology acquired links much faster than its competitors

 Bose-Einstein condensation  At a certain critical temperature, a significant majority of molecules in a gas reach lowest energy state  Prediction could not be proven for 70 years – needed one millionth of a degree of Kelvin  1995 rubidium atoms cooled to form a Bose- Einstein condensate

 Networks can undergo Bose-Einstein condensation  Fittest node can theoretically take all of the links in a network  Single node can exhibit “winner takes all”

 Two network topologies exist:  Scale-free fit-get-rich behavior Most complex networks Fittest node = biggest hub = peaceful competition  Winner takes all behavior Star topology – single hub & tiny nodes No significant competition

 Microsoft Windows  Not scale-free – oldest OS would be most popular  Not fit-get-rich – competitive nodes  Typical competitive market is completely absent  Operating systems = nodes, Users = links  86% of all PCs have Windows

 Summary of Network models  Random networks = random graphs  Scale-free = dynamic with nodes & links  Fitness = competitive nodes fight for links  Bose-Einstein = “winner takes all”  Scale-free is most popular  Web, Internet, Hollywood