Understanding how many can become one

Mathematics has a way of making connections between apparently unrelated human experiences. For example, what’s the common element between a helicopter rescue effort in a war zone, an analysis of how wealth and power consolidate into the hands of the few, and an assessment of whether a single company can achieve a monopoly in a particular market? They are all connected through a mathematical concept called “acquisition,” which is the focus of research being conducted by third-year mathematics PhD candidate Somnath Kundu.

Kundu’s work falls within an area known as Graph Theory which studies interactions between entities in a network or “graph” that have relationships and connections with each other. The individual entities are referred to as “nodes” and the connections between them are “edges.” And when the connections between nodes are determined by some degree of probability, the network is called a “random graph.”

“We study the process by which a node takes on the weight of a neighbouring node,” says Kundu. “In a given graph, there are many possible sequences by which the nodes can be progressively accumulated, leading to a smaller number of nodes. Our research explores the process by which the entire weight of a network accumulates into the smallest possible numbers of nodes.”

By studying acquisition, Kundu’s research may help to answer questions about those disparate scenarios. How can a group of people in a war zone gather in one location to increase their likelihood of being rescued? How can understanding the accumulation of wealth or power inform efforts to build a more equitable economic, political and social system? Or how can acquisition models shed light on ways that markets for certain products and services tend to consolidate into a few, or even one, company?  

“If we think about acquisition in terms of markets, we can see that when markets are distinct from one another, one company will never be able to buy all assets,” explains Kundu. “But, when there is a single market in which all of the entities are connected, one player can typically dominate the market by acquiring neighbouring nodes, provided that an optimal strategy is used.”

In partnership with his supervisors, Dr. Konstantinos Georgiou and Dr. Pawel Pralat in the Department of Mathematics, Kundu recently published a paper titled "The Unit Acquisition Number of Binomial Random Graphs" (external link)  in the Electronic Journal of Combinatorics, which shared novel findings that advance the study of random graphs.

Building on extensive studies of acquisition under deterministic conditions, this research developed tools and skills that allow for analysis of random structures which is, arguably, more informative for real life scenarios. In particular, the research shows that any large, fully connected random graph, which is commonly used as a good model of many real-life situations, can be accumulated into one single node.

The novelty of the work, which is partly related to the ambition of its design, led to an emotional ride for the researchers. “In this particular work, there were many different components which needed to satisfy different criteria if the main results were to hold,” explains Kundu. “Ultimately, we found that all the loose ends were tied up and every different check was mathematically proved to be true. It was a moment of relief when we realized our approach and conjecture were correct, which led to a very interesting and profound result.”

Kundu and his supervisors are planning to build on the current research, which was conducted within the context of the Erdős–Rényi random graph model, by exploring the possibility of studying the unit acquisition process within the context of geometric random graphs, where the underlying “geometry” aims to capture various important aspects of real-world networks, such as locations of companies acquiring assets or interests of users of Instagram or Twitter.

^{Funding for this project was provided by NSERC researcher grants secured by Dr. Georgiou and Dr. Pralat, along with support from Ryerson University and an Ontario Graduate Scholarship secured by Kundu.}