[#49] Synchronization in Power-Law and Erdős-Rényi Graphs with Kuramoto Dynamics: A Graphon Approach & Building a network of mentors

The Kuramoto model is a coupled phase oscillator model that has garnered significant attention over the past decade due to its mathematical tractability and its ability to describe a wide range of phenomena, including circadian rhythms, flashing fireflies, Josephson junction arrays, and high-voltage electric grids. In many of these applications, synchronization is an emergent behavior observed over time and is a key focus of study. The Erdős-Rényi random graph model is perhaps the simplest random graph model. Given n nodes, the probability of an edge existing between any pair of nodes is defined by a fixed probability p. A well-known result for the Erdős-Rényi model is its connectivity threshold: log (n) / n. Specifically, if p is greater than log (n) / n, the probability that the network is connected approaches one as n goes to infinity. Conversely, if p is less than log (n) / n, the probability of connectivity approaches zero as n goes to infinity. In this talk, I will discuss the relationship between synchronization of Kuramoto networks interacting on Erdős-Rényi random graphs and the connectivity threshold, leveraging a mathematical tool called graphons.

When I started my PhD at Cornell, I expected a more “traditional journey” of working closely with one advisor for the next five to six years. Instead, I benefited from a far richer experience: A diverse network of mentors who have shaped my growth. To this end, I would like to propose having a conversation about building a network of mentors. The conversation could possibly focus on the following questions: (1) How can one identify potential mentors beyond their primary advisor, both within and outside their institution? (2) What are effective strategies for cultivating and maintaining meaningful relationships with multiple mentors? (3) How can mentors complement each other in offering diverse perspectives and support, particularly in interdisciplinary research?