January 26, 2004

Expect Unexpected Fluctuations

Fluctuations in Network Dynamics, M. Argollo de Menezes and A.-L. Barabási, Phys. Rev. Lett. 92, 028701 (2004)

Traffic across a complex network, be it packets across internet, signals in a microchip or water through a river network, fluctuates. This paper shows that there is an apparently general patternt o the size of fluctuations compared to the normal flow across a place in the network. Either the fluctuations scale proportionally to the normal flow (examples: the WWW, rivers and highway traffic), or they scale as the square root of the normal flow (Internet traffic, microchip signals).

The paper makes a nice argument for why this is to be expected. In short, networks where the activity fluctuations are driven by internal randomness have fluctuations that are square root sized. Networks where external factors are important (people rushing to news websites when a disaster happens, rivers receiving rain) have fluctuations proportional to the normal flow. In principle there could also be intermediate forms too, especially for smaller networks.

If the argument holds, it has some interesting consequences. Quite often bad things happen when a fluctuation that is large compared to the normal flow happens: Internet routers clog, rivers flood, traffic jams. The max capacity of nodes are often proportional to their normal capacity.

Networks with square root fluctuations have the nice property that the fluctuations get smaller compared to the normal flow (and capacity) for more loaded nodes. Hence it is unlikely that they get overloaded. Good news for the Internet. But networks with fluctuations proportional to normal flow have a fixed risk (if capacity is proportional to the normal flow) per node of failing. Hence we should expect overloading in the WWW and traffic jams here and there.

The risk of failure can be decreased by adding more capacity, but that costs. Cost concerns will tend to reduce the capacity until the cost of failures will equal the marginal cost of capacity (and in the real world, when a major disaster has not happened, usually below this). I wouldn't be surprised if this led directly to a power law distribution of failures.

My guess is that some networks are to be expected to show unexpected fluctuations that cause trouble, even when they function optimally. The systems to look out for are those linked to a changeable, random environment. Thats is, practically all important and interesting systems.

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