social and cultural implications of the discovery of power laws and scale free networks

The social and cultural implications of power laws and scale free networks is these theories can be used to describe how societies or cultures are formed. For example, as the power law suggest societies/communities are in continuous growth. There could be individuals in a community that has only a few acquaintances but others with hundred's of acquaintances. Those with the most connections are likely to be wealthier, since the more connections you have the more likely you will hear about profitable situations. As in scale free networks, people preferentially like to hang out with others who are more popular.

When large pools of connections are made resulting in the formation of a community there is a point where the community becomes too large (i.e. outgrow the amount of physical space) that eventually a part of the community split off and form another network.

Essentially the power law and scale free networks describe how disorder becomes order. The initially disordered population of the world form ordered communities and connections through self-organization behaviour such as preferential attachments.

Societies and cultures are like scale free networks as, they not static and a constantly changing with the times. The connections between people are restructures as individuals change in their personal development. For example, at age 5, your have a certain network of friends, as you grow older your interest changes and the network of friends changes.


power law distribution <------> scale free networks

The power law distribution is a mathematical expression that describes the nature of the World Wide Web. Unlike the behaviour of random networks, which embodies a bell curve distribution, power law distribution implies the characteristic most nodes are not common in the number of links or their characteristics. That is, networks do not have a "characteristic scale in its node connectivity, embodied by the average node" and do not display a peak that the bell curves suggest. In contrast, real network is a continuously decreasing curve "implying that many small events coexist with a few large events". For example, there are many websites on the World Wide web have 5 to 10 links that coexist with a few websites that have 1000's of links.

The power law's relationship to scale free networks is the power law is a mathematical way of describing scale free networks. In a web context, scale free networks have a power law distribution in that there is no single node (i.e. WebPages) that has the characteristic of all WebPages. The nodes of a network follow a power law distribution, since there is a continuous hierarchy of WebPages that have many links, few links to even less links. Moreover, there are only a few websites that are mostly visited. The "power law characterized by a unique exponent, telling us, how many very popular web pages are out there relative to less popular ones".


scale free networks

To define a scale free network the concept of scale free model needs to be explained. The scale free model explains the behaviour of real networks. The scale free network in the context of the Web is essentially WebPages are unique in their nature and connectivity. Thus, there is no scale in which the web is measured from or no WebPages that embodies most of the characteristics of all WebPages. As described in the Rich Get Richer chapter, two laws govern scale free networks: growth and preferential attachment. The development of the Web is a result of one web page linking with other web page sequentially (i.e. one node connecting with another node, and so on) leading to the growth of the World Wide Web. The probability of a person surfing the Web to chose a particular web page is described by the preferential attachment rule. A person is more likely to choose a web page that is the most popular, that is has the most links.

Therefore, the distinguishing features of scale free networks are:

-Network is not static and will grow. A network is build starting from the connection of two nodes.
-The chose of choosing nodes in the network is not random but is preferentially chosen (i.e. new nodes prefer to attach to more connected nodes)

The preferential attachment feature of scale free networks implies not all nodes/WebPages are equally likely to get links. "WebPages with more links are more likely to be linked to again". Moreover, each WebPages is not equivalent to one another in that the older nodes are "richer" since, it has the longest time to collect links. In contrast, "the poorest node is the last one to join the system, with two links only".

This is a brief diagram of my web base social network.........................

small world networks (clustering and 6 degrees of separation and the role of hubs)

The principles of clustering, six degrees of separation are as follows:

Clustering is formation of small groups of individuals/items that are connected. A larger cluster is formed by the connection between the small groups.

Six degrees of separation is the theory that any two individually can be link through the relationship/association of six individuals between them. For example, the connection between two theoretical individual, for our purposes is A1 to B1 can be traced through six individuals. A1 -> A1 sister -> A1 sister's class mate -> Class mate's cousin -> Cousin's brother -> Brother's best friend -> B1

The role of hubs in the formation of small world networks is hubs/connectors facilitate the formation of small world networks. Connectors are "nodes with an anomalously large number of links" (pg 56). The Kevin Bacon game Barabasi writes about is a good illustration, of a connector. Because Kevin Bacon is involvement in many movies he serves as a connection between many actors in Hollywood. If he was did not act, many connections between actors will drastically lengthen creating a bigger world. Kevin Bacon is a connector that shortens the distance/associate between actors, creating a small world network.


regular networks, random networks, and small world networks

My understanding of the definition of regular, random and small world networks is as follows:

Regular networks: are networks where the connection between individuals of the group is direct, that is there is a close relationship between them. For example, my family tree is a regular network; each member of my family can be directly traced to each other family member.

Random networks: is the connection of nodes/clusters/individuals through indirect relationships or random encounters. Each individual of a random network can obtain information from other individuals of the network through random encounters. For example, a network can be formed at a cocktail party of strangers when these individuals communicate with one another, where eventually links between individuals are formed resulting in a network. Erdo¡¦s wine party analogy is a good illustration of this.

Small world networks: highly interconnected network of individuals with something in common. Moreover, small world networks are characterized by small distance between nodes. For example, a network of scientist represents a small world network. Small world networks can arise from random or regular networks. The reason why on average individuals knowledge of one another can be traced through six individuals is "humans have an inborn desire to form cliques and clusters that offer familiarity, safety, and intimacy" (pg 50)


Random Graphs 2 small world networks

The history of how we think of network links today originated in 1736 from Euler¡¦s solution to the Konigsberb problem (i.e. Can one walk across the seven bridges of Konigsberg an never cross the same one twice?) by way of graph theory. In Euler¡¦s graph he illustrated the seven bridges pictorially and used nodes (i.e. destination points) and links (i.e lines) connecting the nodes, which determined how many possible paths, there were between destination points.

In 1950's two Hungarian mathematicians, Paul Eros and Alfred Renyi made a revolution in graph theory by addressing how do networks form? A cocktail party best describes their theory. The question is if one person informs another stranger at the party about a wine will all of the people of the party know about this wine by the end of the party. Erdos and Renyi illustrated the encounters of the guest of the party by creating a graph. Nodes represent the guests and each social encounter creates links that connect the nodes, this illustration results in a graph. Erdos and Renyi goes further by adding the concept of randomness to replicate real life. By creating random links between people at the party they found that eventually everyone was connected in some way. Their random graph theory has been used to explain how real world networks are formed.

The nature of real world networks and our understanding of them can be best summarized by the following quote "The construction and structure of graphs or networks is the key to understanding the complex world around us. Small changes in the topology, affecting only a few of the nodes or links, can open hidden doors, allowing new possibilities to emerge." (pg 12)

Sean