Mathematician devises equations to untangle 'controlled chaos'
"By providing reliable results in a rapid manner, these equations allow for the creation of algorithms that optimise the resilience of real interdependent networks," said the study's author, Filippo Radicchi, of Indiana University, whose work appears in the journal Nature Physics.
"They may also be helpful in designing complex systems that are more robust, or more easily recoverable," he said.
Radicchi's equations provide a new method to "untangle" multiple complex systems – pulling apart each network or "graph" for individual analysis and then reconstructing an overall picture.
A "graph" describes the myriad points and connection lines that comprise a complex network. In an air transportation network, for example, an airport might represent a single point; an airplane's flight path, the connections between points.
"In the real world, networks do not exist in isolation; they are always interacting with other networks," Radicchi said. "By unraveling multiple graphs, we're able to analyse each in isolation, providing a more complete picture of their interdependence and interaction."
The equations are able to quickly and accurately measure "percolation" in a system – the amount of disruption caused by small breakdowns in a large system.
"If you're traveling between cities by plane and 10% of the airports worldwide suddenly stop operating for some reason, percolation theory can help us calculate how many airports you can still use to reach your target city," Radicchi said.
A smooth percolation transition, as revealed though the equations, indicates that a system will stop functioning gradually as the number of local failures rise. An abrupt percolation transition reveals a system more likely to stop functioning suddenly after reaching a certain number of local failures.
"At that point," Radicchi said, "a system will exhibit 'catastrophic behavior,' from which it is very difficult to recover."