A small world network is a complex network made up of nodes and links.  A node could be a person, a place, a computer or even an atom. The nodes need to be sufficiently autonomous to allow the diversity of qualities within the network to interact. The autonomy allows them to interact and build connectivity with the other nodes through the links they have.

If a network has too few links there is not enough information flowing for the network to operate effectively, but equally, if there are too many connections, the system becomes clogged with too much information flowing. An example of a clogged network would be an organisation which is top heavy with bureaucracy. So much time is taken maintaining the bureaucracy that the network does not get on with  its actual function.

Duncan Watts investigated different types of network to find which were more effective. He looked at a network where the nodes were linked in a circle so each node is only linked to the node on either side. This is obviously not very effective as many steps are required to send a message from one side of the ring to the other.

At the other end of the scale is a random network. This also turns out to not be very effective. He interestingly found, however, that if we take just one percent of the links in the circular network and reassign them at random around the network, the network makes a sudden leap in effectiveness. It only takes a few "shortcuts" to dramatically reduce the number of links on the network to anywhere else on the network.

If we look at small world networks in the real world we will find some nodes are heavily linked while others have very few links. Some people seem to know everyone, while others are isolated having little contact with other humans.

We tend to have a cluster of people we are closest to, who know is very well. These people are likely to be our greatest support when we need it.  The other people in our cluster will tend to know each other well and have good knowledge and skills in the same areas as ourselves.

Mark Granovetter investigated the importance of weak links. These are the people we do not know as well. They are people we know enough to make contact with if we have a particular reason.

If I wanted to shift to another city and wanted to find out all about it or wanted information about an unusual topic, it is not that likely that those in my own cluster will be able to help me. Those people within my weak networks  are more likely to be able to help me. Somebody such as a person I met on a training course two years ago, who lives in the city I want to shift to will be a very useful contact and the links I can make within his or her cluster will be extremely useful. Each weak link can be the entry point to a whole new cluster.

As mentioned, there are people who have many more links than those around them. They seem to know everyone and are connected to a large number of clusters. These people are hubs of the network. They are the key people that hold the network together and give it strength and flexibility.

Small world networks are very strong against a random attack because the number of more critical hubs is usually very small compared with the entire number of nodes. It is far more likely that a random attack would affect a node that is not greatly connected to the network and the network can find other ways; using other nodes and links to ensure that any necessary functions can be maintained.

A small world network is vulnerable to a targeted attack. If the attacker can identify a hub and attack it, it will have a significant impact on the ability of the network to recover. In the really world, hubs tend to be heavily defended. The 9/11 terrorists were able to have such an effect because they chose a hub target.

The city of Dunedin, where I live, has a population of around 100,000 people. Imagine that normal communication systems were not functioning and we wanted to set up an efficient telephone tree, so information could be relayed to every citizen as quickly and efficiently as possible. How would we set it up?

First, we would need to decide how many people each person in the tree was to contact.

Two calls per person	Total contacted	Ten calls per person	Total contacted	25 calls per person	Total contacted
1	1	1	1	1	1
2	3	10	11	25	26
4	7	100	111	625	651
8	15	1000	1111	15,625	16,276
16	31	10,000	11,111	390,625	406,901
32	63	100,000	111,111
64	127
128	255
256	511
512	1,023
1024	2047
2048	4095
4096	8191
8192	16,383
16,384	32,767
32,768	65,535
65,536	131,071
131,072

If we decided each person would phone two others, then we see from the chart that we would need 17 layers of phone calls before everyone was contacted. If instead of contacting two people, each person had to telephone 10 others, then only six layers would be needed, and if each person had to telephone 25 others, only five layers would be needed.

So, what is the best, phoning two people, 10 people or 25 people?

Telephoning only two people is likely to be slow because of the number of levels and with the large number of levels, the likelihood of errors and lost messages increases. Telephoning 25 people is also not likely to be efficient, because each layer would take so long to complete. Adding only one more layer means that each person only has to phone ten people, which is much more manageable.

So, of the options above, 10 is more likely to be the most efficient number for each person to contact.

Another alternative would be to vary the number of people contacted at the various levels. So, we might have people in levels 1-3 phoning 10 people, those from level 4 on phoning 2. I think you can see that we would create a bottleneck that would slow the system down. That would also be the case if 25 people were contact from level 4 on. It would not be as efficient. It turns out to that the most efficient way of getting the message out is to find the right number of people for each person to contact and to keep that number constant for all levels.

 A tree has the same basic problem as a person setting up a telephone tree. Instead of sending out a message to all the people in Dunedin, a tree needs to send nutrients up the trunk to be evenly distributed to all the leaves. Nature has found the most effective way. It starts with a single large trunk, which then divides into smaller branches, say half the size of the trunk, which in turn divide into branches half the size again, then half the size again. Each time, the branch is reduced by half the size until you reach tiny little twigs. Nature uses the same equal proportions between levels because that is what works best. The size of the branches decreases by the same proportion at each level. If we changed the proportions as we talked about in the telephone tree so there were big branches all a sudden far from the trunk, or had lots of small branches coming straight off the trunk, it would not be as efficient.

The most efficient way of getting all the cars in a large flow of traffic entering a city to quickly reach its destination out in the suburbs is again to use equal proportions. The best way is to have one or a few large motorways coming into the city on which cars travel very fast, then branching off into smaller main roads, which in turn branch off into smaller roads leading to the suburbs, where there is less traffic, but it moves slower.

Again, if we had many small roads coming straight off the motorway, or large main roads through the suburbs traffic flow would be decreased and there would be many traffic accidents because of traffic jams caused by cars slowing right down to turn off a main highway.

There are so many other systems we come across in our daily life that fit this pattern. Our lungs breathe in air down one big tube, which is then distributed through smaller and smaller branches to the tiny alveoli so oxygen can enter our blood stream. Our circulatory system moves blood from the heart to the tiny capillaries so the blood gets to all the parts of the body. Our telephone system sends voice messages, our power grid sends electricity, the postal system sends mail, our body sends oxygen and blood. We have different systems, but each time equal proportions between levels is the most efficient way of organising them all.

We can see that the same system of proportions also works, when we turn it in reverse. People taking their cars out of the city need roads that evenly increase in size instead of decreasing as before. Small streams combine into larger and larger rivers until they meet in one main river flowing to the sea. In these systems the proportions increase equally at each level rather than decrease.

Airlines move passengers all around the world. What is the most efficient structure linking the airports? The most effective way of setting up an airport system is to have smaller airports feeding into bigger airports. Once the biggest airport between the starting point and destination is reached, the process is reversed and we switch to smaller and smaller airports until we reach our destination. If I want to travel from Dunedin in New Zealand to Santa Fe in New Mexico, I fly first from the smaller airports to the to the larger, Dunedin to Auckland to Los Angeles, then on to smaller airports to Albuquerque and on to Santa Fe. It would not be efficient to have flights direct from Los Angeles to Santa Fe or even Los Angeles to Dunedin. Large airports like Los Angeles are hubs that are crucial to the structure of the whole network. Airport sizes are shown to have equal proportions between levels.

 

If we measure the number of earthquakes of every size you find the same pattern yet again. If there is one earthquake at size eight and if the proportion between layers is ten, we would expect around 10 earthquakes at size seven (The proportion is not ten but about 2.7, but using ten makes it much easier to understand the pattern). At size six earthquake we would expect about 100 earthquakes and at size five earthquake we would expect around 1,000 earthquakes. This pattern continues right down to 10 million earthquakes at size one. As with any natural system, the numbers do not match the perfect mathematical pattern exactly, but in real life it is accurate enough to show the relationships exist.

The same pattern of proportions is found in the nerve cells of our brain. There are literally billions of nerve cells in our brain. A small number of nerve cells are linked to many thousands of other cells, while a large number of others are only linked to a small number of other cells. When you examine the proportions, we find the same equal proportions between the numbers of connections.

These proportions are used to help diagnose diabetes. One of the first signs of diabetes is irregularity in the blood vessels in the eye. If the proportions of size of blood vessels in the eye fit the patterns discussed here, it is likely that the person does not have diabetes. If there are proportionally more smaller blood vessels or larger blood vessels, breaking the proportions, then it is likely that the person has diabetes. This is a screening method and not as accurate as more full tests, but this method of diagnoses is much cheaper than standard methods and can be very useful especially in places where there is no equipment for the full test.

An investigation of our social connections shows that they also exhibit equal proportions between levels.  We have a smaller number of family members or very close friends who know us well. The next layer has more friends we know well and might see every few days or each week. As we look at succeeding layers we find more and more people, whom we know less and less. On the outside are a lot of people we know only a little.

You might think it is not important to have a large number of people we only know through weak links. Imagine, however, that I decided to go and live in Wellington. My friends and family are unlikely to be able to give me the information and advice I seek. It is far more likely, for example, that people I met on a training course last year who live in Wellington, will be much more use to me. I don’t know them at all well, but well enough to telephone. They will probably also know other people in Wellington to answer questions they cannot answer. Having weak links with a wide range of people, who each have their own areas of interest, networks, and resources can be extremely important. When we measure social connections, we find equal proportions at each level of closeness because this proves to be the most effective mix to meet all our social needs.

We have found equal proportions between levels in telephone trees, trees, roadways, electricity grids, earthquakes, sand piles, social connections, lungs, hearts, brain cells, rivers, airports and postal systems, but, it does not end there! The very same equal proportions between levels can be found in the shape of clouds, the size of traffic jams, the size of towns and cities people live in, the amount of money in people’s bank accounts, the number of people killed in wars and even the number of sexual partners we have in a lifetime.

So, while we all make our own decisions about our lives, somehow together we make choices that maintain the same mathematical patterns of equal proportions between levels.

This picture looks like it could be a plant or a brain. In fact, it is a map of the internet. Why does a map of the internet look so organic? It looks organic because it has the same underlying mathematical proportions as a living form. That is particularly interesting because the internet is a distributed network with nobody in charge. With literally many millions of people each acting autonomously, at first glance we would expect an enormous random mess, whereas we find a highly organized, dynamic structure. A few sites have millions of links to them, while the vast majority only have a handful. It probably won’t surprise you by now to find the links form equal proportions between levels. At this point we start to ask the question: to what degree can we call the Internet a living being? It is stable, creates and sustains flows of energy as information like a body sustains flows of energy as food, blood, etc. It gets viruses and has an immune system, parts die off while new parts come to life and grow and it has exactly the same underlying mathematical blueprint of equal proportions between levels that all life has. Is it like they say on Star Trek, “life, Jim but not as we know it? Where exactly is the boundary between what is living and what is not? We can ask the same question of the city of Dunedin, the road transport system, the postal service, the banking system, the stock market and other things we would clearly call, “man made”. What does this mean for our usual definition of life.