How Much Data Does the Theory of Severity Take to New Dimensions

Graphic theory is not igo ra

The mathematical language for talking about connections, which often relies on networks – lines (dots) and edges (lines that connect them) – has been an important way of modeling real -world events since at least in the 18th century. But a few decades ago, the emergence of giant data sets forced researchers to expand their toolboxes and, at the same time, gave them many sandboxes in which to apply new mathematical knowledge. Since then, it is said Josh Grochow, a computer scientist at the University of Colorado, Boulder, is having an exciting period of rapid growth as researchers have created new varieties of network models that detect complex structures and signals in noise in a lot of data.

Grochow is with a growing chorus of researchers who point out that when it comes to finding connections across multiple data, graph theory has its limitations. A graphic represents each relationship as a dyad, or pair interaction. However, many complex systems cannot be represented by binary connections alone. The current development of the field shows how progress is being made.

Consider trying to forge a parenting network model. Obviously, every parent has a connection to a child, but the parenting relationship is not just the summation of the two links, because the graphic model can model it. It’s the same with trying to model an event like peer pressure.

“There are a lot of intuitive models. The effect of peer pressure on social dynamics can only be obtained if you have groups in your data,” he said. Leonie Neuhauser at RWTH Aachen University in Germany. But binary networks do not capture group influences.

Mathematicians and computer scientists use the term “higher order interactions” to describe the complex ways in which group dynamics, rather than binary links, can influence individual behavior. These mathematical wonders can be seen in everything from the involvement of interactions with mechanics as a whole to the course of a disease that has spread to a population. If a pharmacologist wants to model drug association, for example, graphic theory might show how two drugs respond to each other – but what about the three? Or four?

While the tools for exploring these interactions are not new, it is only in recent years that high-dimensional data sets have become a machine for discovery, providing mathematician and network theorist come up with new ideas. These efforts have yielded interesting results regarding the limitations of the graphs and the possibilities for uplift.

“Now we know that the network is just a shadow of the matter,” Grochow said. If a data set has a complex underlying structure, then modeling it as a graphic can only reveal a limited outline of the whole story.

Emilie Purvine of the Pacific Northwest National Laboratory is excited about the power of tools like hypergraphs to map subtler connections between data points.

Photo: Andrea Starr / Pacific Northwest National Laboratory

“We realized that the data structures we used to study objects, from a mathematical perspective, were less relevant to what we saw in the data,” the mathematician said. Emilie Purvine at the Pacific Northwest National Laboratory.

Which is why mathematicians, computer scientists, and other researchers are increasingly focusing on ways to generalize graph theory-in many of its forms-in order to examine much higher order events. The last few years have brought a flurry of proposed ways to identify these interactions, and to validate this mathematics in large-scale data sets.

For Purvine, mathematical analysis of higher-order conversations is like mapping new dimensions. “Think of a graphic as a foundation of two -dimensional land,” he says. Three-dimensional buildings that can be built in height can vary. “If you’re at ground level, they look the same, but you’re built differently on height.”

Enter the Hypergraph

The search for higher -dimensional structures where the math turns is particularly dubious – and interesting. For example, the higher order analogue of a graphic is called a hypergraph, and instead of edges, it has “hyperedges.” It can connect to multiple nodes, which means it can represent multiple relationships (or multilinear) relationships. Instead of a line, a hyperedge can be seen as a thread, like a tar that is placed in three or more places.

Which is good, but much more we don’t know how these structures relate to their usual counterparts. Now mathematicians know what rules of graph theory are also used for higher order interactions, suggesting new areas of exploration.

To illustrate the different relationships that can be gleaned from a hypergraph from a large set of data – and not from an ordinary graph – Purvine points to a simple example close to home, the world of science publication. Imagine two sets of data, each with internal papers written in three mathematics; for simplicity, let’s name them A, B, and C. A data set contains six roles, with two roles in each of three different pairs (AB, AC, and BC). ). Some contain only two papers, each written in all three mathematics (ABC).

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