Feb. 28 (UPI) -- A new theorem designed by mathematicians from Carnegie Mellon University and the University of Pittsburgh suggests Pennsylvania's congressional districts were gerrymandered.
The theorem is based on Markov chains, algorithms used to model how a fixed object can evolve, through incremental and linear changes, into a different but random shape.
Researchers have previously used Markov chains to simulate thermodynamic processes, chemical reactions, economic phenomena, DNA sequencing and more. But Markov chains aren't perfect. Some have criticized the modeling theorem because it's not clear how long the simulation must be allowed to run its course before a sample represents a truly random result.
For the new study, published in the journal PNAS, mathematicians at the University of Pittsburgh and Carnegie Mellon University developed a way to use Markov chains to determine whether a sample is non-random without producing any random samples through simulation.
Researchers used their new method to analyze Pennsylvania's congressional districts map. With the current map as the starting point, researchers used the Markov chain to randomly alter the congressional districts. The chain's incremental changes were governed by a few restrictions. Changes had to ensure equal populations in each district, maintain border continuity and limit the ratio of border perimeter to total area.
The random incremental steps of the map's evolution resulted in large statistical changes -- a phenomenon that would be highly unlikely had the map been produced free of purposeful political bias.
"There is no way that this map could have been produced by an unbiased process," Carnegie Mellon researcher Wesley Pegden said in a press release.
The problem of gerrymandering has inspired a variety of mathematicians. Moon Duchin, a professor of mathematics at Tufts, trains other mathematicians in mathematical concepts relative to gerrymandering. Duchin uses geometry to identify problematic districts. Specifically, Duchin uses metric geometry theorems to measure the "compactness" of congressional district shapes.