May-29-2017

Chapter 3: Six Degrees of Separation

Six Degrees of Separation was first published by Karinthy in 1929 and then rediscovered by Stanley Milgram in 1967.

Everybody on this planet is separated by only six other people. Six degrees of separtion, between us and everybody else on Earth. 

Page 34: Small separations are common in just about every networks, e.g.,

(1) The Web, with a billion nodes, has a separtion of nineteen degrees (i.e., diameter of the web is 18.59).

(2) The internet, with hundreds of thoudsands of routers, has a separtion of ten. 

(3) Species in the food webs are on average two links away from each other;

(4) Scientist in different fields of science are separated by four to six coauthorship links. 

Why, how do networks achieve such a uniformly short path despite consisting of billions of nodes?

A network, the average links for nodes are: k links. This means from a typical node we can reach k other node with one step. Hence, there are k^2 nodes two links away and k^d nodes exactly d links away. Therefore, if k is large, for even small values of d the number of nodes you can reach are huge. Within a few steps your have reached all nodes to be found, which explains why the average separation is so short in most networks. 

If we have N nodes in a network, k^d must not exceed N. Thus, using:

k^d = N

The average separation follows the equation:

d= log N / log k

Chapter 5: Hubs and Connectors

Malcolm Gladwell

Page 56: Connectors——nodes with an anomalously large number of links——are present in very diverse complex systems, ranging from the economy to the cell.

Page 58: Hubs——a few very highly connected nodes, with large number of incoming and outgoing links. Remove a few of them will cause the shortest distance between nodes in the network drastically lengthen.

Page 61: As networks are clustered, nodes that are linked only to nodes in their cluster could have a central role in that subculture or genre. Without links connecting them to the outiside world, they can be quite far from nodes in other clusters. (e.g., the porn stars). The truly central position in networks is reserved for those nodes that are simultaneously part of many large clusters. 

Page 64: Useful for writing: Ecologists believe that the hubs of food webs are the keystone species of an ecosystem, paramount in maintaining the ecosystem's stability.

(paramount: to be paramount / to be of paramount importance是至关重要的)

Page 64: Hubs are special and deserve attention. They dominate the structure of all networks in which they are present, making them look lile small world.

With links to an unusally large number of nodes, hubs create short paths between any two nodes in the system. Consequently, while the average separtion between two randomly selected people on Earth is six, the distance between anybody and a connector is often only one or two. 

Questions remaining to explore:

How do hubs appear?

How many of them are expected in a given network?