Tipping Points

Stuart Kauffman devised a mathematical model which he explains with the analogy of a pile of buttons on a table. At random, the individual button are tied to another button. Each tie forms a link. So for a long while we still have a collection of buttons with sporadic links between some. Over time, however, connected buttons will get connected to other buttons, so some buttons will be densely connected to other, while most will remain with one or very few connections. The pattern of links will fit the equal proportions as discussed in the small world network page with some buttons densely connected and most with very few..

If we now see that button system as a community of people and the ties are the communication links between them, we find, as we would expect, that at the start, when only a few of them are connected together, that the communications and overall effectiveness is very low. The more links we have the more efficiently the community can communicate and get on.
The community will also grow to have a small number of people who are richly connected and the greater majority only having a few or even no links.

Interestingly, we find is that as the number of links nears the number of people in the community there is a sudden increase in the effectiveness of the communications of he community. Then as more and more links are made the community reaches a saturation point in links. People become overwhelmed by the number of link they have to keep up. Now, the addition of more links adds less and less extra effectiveness to the community.

If we graph the number of connections within the community against the effectiveness of the community, we get the graph in the diagram  to the left. You will note the small range marked on the graph where only a few more connections drastic increases the effectiveness of the system. This is called the 'tipping point'.

We find tipping points  in many real world situations. The internet grew slowly in the first few years then quite rapidly suddenly there were enough webpages and connections to make it worthwhile people joining in and then it took off.

There are also examples in our world where we want to avoid the tipping point. The spread of a virus through a population moves very slowly when only a few people are affected and the virus but if a critical number of connections are made, then the virus, be it AIDS, Avian bird flu, or just the common cold, can suddenly turn into an epidemic. The aim is therefore to reduce the connections to keep the system away from the tipping point.

The national grid is a complex system of interlinked power stations. If one substation overloads it puts pressure on the surrounding sub-stations to cope making it more likely that another sub-station overloads. Depending on the exact nature of the system at the time, a chain reaction may take place knocking out a significant portion of the whole grid. This is how the enormous black-outs occurred in the United States affect cities the size of New York. So, a sub-station  failure might have no effect on the network, or may bring it to its knees and because the network is so complex, we cannot tell beforehand, what effect will result.

There are several factors which will tend to take a system towards or away from the tipping point. First of all is the number of connections between the agents,  the stickability. In the example of the virus, it would have a greater stickability if the virus remains active for longer enabling more people to be infected from the one person.

If some links are more highly linked than others, they will be more critical to the functioning of the network. If they fail to be able to function, the effect on the whole system can be severe.

The Law of Context is about the environment the network exists within. A virus might spread much more quickly in certain conditions like high temperatures, or a new product might reach the tipping point much quicker if the economy is more buoyant, or the item suddenly becomes more fashionable.  The nature of the agents in the network also affects the ability to operate.  Human beings have some inbuilt tendencies. Our short term memory can hold about seven pieces if information. We can maintain relationships with around 150 people, before we start to get overwhelmed. So if the network is organised in groups of 150 people and information is present in chunks of seven pieces, then the network will operate more effectively. Whenever human groups grow beyond 150, they tend to break into two groups.

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