It has been observed time and time again in physics, biology, economics, sociology and anthropology that a small number of the nodes within a network often form hubs, each of which possess more links than the rest. Some of these hubs have a substantially higher number of connections than their counterparts. The distribution of the connectivity of nodes in such networks commonly follows what’s called a “power law”, with a small number of super-hubs, a larger number of more modestly connected hubs, and the rest; nodes possessing just a few connections.

If you are an actor one link away from Rod Steiger, you are connected by just one additional link to everyone in Steiger’s network. Steiger is what network theory would call a hub: he is very well connected, possessing what is known as a high degree of “centrality”. Here are two other examples of hubs in networks.

Just over 2000 years ago, a poor carpenter died in an obscure backwater of the Roman Empire. Most of his followers were Jewish, persecuted by their Roman rulers, and few in number. Their creed was unpopular with non-Jews, for several reasons: the fact that to join their religion men had to be circumcised was significant. A wealthy Roman citizen called Saul was converted to the cause. He changed his name to Paul, modified the way this new religion was practised, removed the need for circumcision, and merged into it ideas from the Greco-Roman world. Using his extraordinary network of contacts, he preached his newfound religion in terms his fellow educated Romans and Greeks could understand. Paul was a hub: a highly connected node in the network that was the Roman world.

Gaëtan Dugas was a good-looking Canadian flight attendant. His job entailed him travelling large distances. He was also extremely sexually active within the gay community, claiming to have had over 2500 sexual partners. Unfortunately, he also contracted AIDS. Before dying of kidney failure in 1984, Dugas is thought to have spread AIDS to many hundreds of people and is described as being “patient zero”, a central hub in the early spread of the disease in North America. Had Dugas not contracted the illness its spread might have been far slower, giving scientists more time to study it and giving the message of safe sex more time to become widespread. On the other hand, it was inevitable that a hub such as Dugas would be among the first to catch the disease. It is likely that there were not many degrees of separation between the hubs in the network of AIDS sufferers and more peripheral figures that may have contracted the disease earlier.

According to studies by Albert-László Barabási and colleagues at the University of Notre Dame, hubs appear with remarkable constancy in networks ranging from the spread of diseases, the make-up of ecosystems, cells within animals and plants, the economy, and across the Internet, both at the router level or in the Web. For example, in the context of the Web, Google is an exceptionally large hub. It is these hubs that support small world networks: it’s a small world because hubs reduce the number of links between one node and another.

Barabási also found that the occurrence of these hubs tends to follow a power law: of a large number of nodes, there are only a small number of hubs, and a smaller number still of “super-hubs”.

number of us possess 100 or so “friends”. Within that number, there is probably one linked to several thousand. Furthermore, certain “super-hubs” within Facebook may have tens of thousands, even hundreds of thousands of connections. Barabási suggests that a number of factors need to be in a place for such hubs to emerge. One of these factors is growth. In a growing network, one would expect nodes that have been in existence for longer to have more links.

Such networks are described as being “scale-free”.

Another important factor is what is known as preferential attachment. We all intuitively understand the principle of preferential attachment. Imagine two similar cafés opposite each other on the street. The first customer of the day has no preference; he randomly chooses one. The second customer may also be unsure, but is more likely to opt for the café which isn’t empty. So, each new customer chooses the same café, until eventually one of the establishments is heaving with customers, the other deserted. This is described as “the rich get richer”. Popular nodes may attract links, their very popularity giving them their appeal.

© Investment & Business News 2013

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