When discussing network theories, one name came up once and again -- Paul Erdős. Paul Erdős is a genius who wrote more than 1500 mathematics paper, more than any other mathematician in history. In fact, Erdős was one of the most prolific publishers of papers in mathematical history, second only to Leonhard Euler.
The first networking relevance this brings is -- The Erdős number
His friends created the Erdős number as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8.
Alfréd Rényi joint papers with Paul Erdős, introducing the Erdős–Rényi model of random graphs. In 1959, a complex network was described as random. But Erdős agreed that real networks must have organizing principles that distinguish them from the random network introduced, though the ideas dominated on network modeling.
The next idea is "Six Degrees of Separation"
Six degrees of separation refers to the idea that, if a person is one step away from each person they know and two steps away from each person who is known by one of the people they know, then everyone is at most six steps away from any other person on Earth.
Today, with air travel and the connectedness of the Internet, people are more connected than ever before. Even when when 2 people are continents apart, it would not be hard today to somehow find a way to connect these people. Six degrees is the product of our modern society -- a result of our insistence on keeping in touch.
Small World
Small worlds are generic property of networks in general. Our ability to connect is reduced through distance and discovering acquaintances with strangers on worldwide trips reminds us that some people on the other side of the planet are often closer along the social network than our neighbors next door. A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. A small world network, where nodes represent people and edges connect people that know each other, captures the small world phenomenon of strangers being linked by a mutual acquaintance.
Everyone has strong ties and weak ties. Weak ties play a crucial role in our ability to communicate with the outside world. Often our close friends can offer little help in finding a job as they move in the same circles and are exposed to the same information. To get new information, we need to activate our weaker ties.
The weak ties or acquaintances are our bridge to the outside world, since by frequenting different places, they obtain their information from different sources than our immediate friends.
Clustering
Humans tend cluster intuitively. We have an inborn desire to form cliques and clusters that offer familiarity, safety and intimacy. The discovery that clustering is ubiquitous has rapidly elevated it from a unique feature of society to a generic property of complex networks and posed the first serious challenge to view that real networks are fundamentally random.
I've got more information and papers to read and catch up and will post more in the next few days.
-- Robin
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