Research Projects

From ecology to neuroscience, recent developments in stability analysis of large networks are essential for comprehending interconnected systems' resilience and long-term behaviour. The question that ecologists are trying to answer is: What components of ecological networks serve as building blocks for biodiversity maintenance? Similar questions might be posed in neuroscience: How can neural networks transmit neuronal signals from one layer to another? One may wonder how minute traits, such as connection patterns, contribute to reliable and stable information processing? Robert May's groundbreaking work, which used random matrix theory to evaluate the stability by looking at the eigenvalue distribution of Jacobian matrices, established this field of study in an ecological setting. Extensive research on stability across many kinds of natural systems spanning complex networks was spurred by May's findings that random interactions can stabilize or destabilize a natural systems. The idea that interactions are entirely random is contested by most contemporary research, which contends that particular dynamical laws govern many real networks. Since standard eigenvalue analyses based on random matrix theory frequently fail to represent the complex dynamics within systems driven by non-random elements, this change has added considerable complexity to the modelling of network behaviour. Thus, despite these developments, the stability of such systems, primarily through eigenvalue analysis, still requires improvement in a broad analytical framework. For example, several questions still require attention: Which node or nodes in a particular graph are in charge of stability? Does it rely only on the intricate nonlinear dynamics? Or do the nodal degrees' higher moments contribute significantly to stability? Second, previous research on neural networks assumed complete or low-rank connection. On the other hand, cortical networks in the real world are typically sparse. Significant problems concerning the consequences of structural heterogeneity are brought up by this study, including how the stability of neural networks is affected by various network features that nodes of varying degrees may represent. As these fields develop, it highlights the necessity of sophisticated mathematical techniques to manage the subtleties of dynamics-based, heterogeneous, and non-random interactions. Understanding bifurcations—critical moments where systems qualitatively change between stable and unstable states—and investigating how structural heterogeneity affects stability in real-world networks are two of the leading research problems.