When working on complex systems and trying to map them you will find that the relations within your data can become very complex very fast. Complex data which is time-dependent and interrelated will form a complex maze of data and dependencies which will become almost a impossible task of mapping.
A group of researchers from Darmouth have developed a mathematical tool which can help understand complex data systems like the votes of legislators over their careers, second-by-second activity of the stock market, or levels of oxygenated blood flow in the brain.
“With respect to the equities market we created a map that illustrated a generalized notion of sector and industry, as well as the interactions between them, reflecting the different levels of capital flow, among and between companies, industries, sectors, and so forth,” says Rockmore, the John G. Kemeny Parents Professor of Mathematics and a professor of computer science. “In fact, it is this idea of flow, be it capital, oxygenated blood, or political orientation, that we are capturing.”
Capturing patterns in this so-called ‘flow’ is important to understand the subtle interdependencies among the different components of a complex system. The researchers use the mathematics of a subject called spectral analysis, which is often used to model heat flow on different kinds of geometric surfaces, to analyze the network of correlations. This is combined with statistical learning tools to produce the Partition Decoupling Method (PDM). The PDM discovers regions where the flow circulates more than would be expected at random, collapsing these regions and then creating new networks of sectors as well as residual networks. The result effectively zooms in to obtain detailed analysis of the interrelations as well as zooms out to view the coarse-scale flow at a distance."
Source Press Release
In a paper named "Topological structures in the equities market network" written by Gregory Leibon, Scott D. Pauls, Danile Rockmore and Robert Savell the Partition Decoupling Method is used to map the underlying structure of the equities market network.
"We present a new method for the decomposition of complex systems given a correlation network structure which yields scale-dependent geometric information — which in turn provides a multiscale decomposition of the underlying data elements. The PDM generalizes traditional multi-scalar clustering methods by exposing multiple partitions of clustered entities. "
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