By: Eric Havens, Sanusha Matthews, and Mike Copciac

RICH GET RICHER

OLD NETWORK ASSUMPTIONS

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.

GROWTH: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.

RANDOM NETWORK VS. PREFERENTIAL ATTACHMENT

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)

THE BIRTH OF A SCALE-FREE NETWORK

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.

THE 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