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Rich-Get-Richer or Power Laws

Third, an empirical fact that corroborates this notion of copying is the existence of the Power Law or the rich-get-richer phenomenon. For example, the fraction of webpages with exactly k in-links is monotonically decreasing in a non-linear fashion and is proportional to 8

In networks, the Power Law best describes the formation of hubs, or nodes with many in-links. The dynamics of hub formation has three stages. First, there is growth of the network, as new nodes are being added over time. Second, these nodes link to senior ones via an algorithm called preferential attachment. Basically, a new node links to hubs or nodes with a large number of in-links. The more in-links a senior node has, the more nodes will preferentially link to this node. This dynamic is called the “rich-get-richer” rule, which predicts that senior nodes will become hubs conferring some sort of a first-mover advantage. In some markets, however, we do observe latecomers frequently outperforming first-movers. This leapfrogging of newcomers may be due to the quality of the individual nodes, as explained below.

Third, there is the notion of fitness of nodes. The assumption that all nodes are equal is clearly not valid. This feature is captured by “fitness,” defined as “a quantitative measure of a node’s ability to stay in front of the competition.” Then “a simple way to incorporate fitness into the scale-free model is to assume that preferential attachment is driven by the product of the node’s fitness and the number of links it has” [96]. Fitness captures the notion of speed such that a node with high fitness grabs links faster than nodes with the same seniority but lower fitness. For example, Google, whose search engine is superior, acquired users much faster than Yahoo.

The phenomenon of hubs has implications for the survival of the Internet. The architecture of Internet is such that for the entire network to collapse, multiple interconnected hubs must be destroyed simultaneously. If the probability of a node being attacked or otherwise rendered dysfunctional is equal to p and is the same, but independent, across all nodes, then the probability of multiple nodes being destroyed simultaneously can be shown to be small. For example if there are n critical hubs, then for the entire network to crash we need all n nodes to break apart - the probability of this occurrence is pn, which tends to zero as n goes to infinity since 0 < p < 1. The Achilles’ heel is the appearance of an informed hacker who selectively and systematically targets critical hubs and brings the entire network to a halt.

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