# Large Time Behaviour and Metastability in Networked Systems

### Sarath A Y

**Research Summary**

In this research, our goal is to understand the large time behaviour and metastability in networked systems such as WiFi networks, cloud computing systems, societal networks with rational agents, etc. We consider a mean-field interacting particle system with N particles. Each particle has a state associated with it which evolves in a Markovian fashion where the rates of transition depend on the other particles only through the empirical measure of the states of all the particles. Such a particle system is useful in modelling many networked systems. It turns out that the performance of these systems can be understood from the stationary behaviour of the empirical measure process, and our research focuses on studying the same. We are particularly interested in studying metastable phenomena such (i) the mean time for the system to be close to stationarity, (ii) the mean exit time from an operating point, etc. We can then use insights from this study to drive better design of such systems. We mainly use the theory of large deviations as a tool to study such questions. A more detailed break-up of my research during the past academic year is as follows.

August 2019 – November 2019: We worked on an extension of the above mean-field model where the system is subject to a fast varying random environment. We established the process level large deviation principle (LDP) for such a system and used this to study metastability phenomena in such systems. This work has been written up and it is under review. November 2019 – March 2020: I visited Netradyne Technologies Pvt. Ltd., where I worked on the impact of real time feedback on driving behaviour in collaboration with Bengaluru Metropolitan Transport Corporation (BMTC). I worked under the guidance of Prof. Rajesh Sundaresan, along with my labmates Nihesh and Karthik. In this work, based on visual data analytics, we proposed an automated approach to identify violations on the Bus Priority Lane (BPL) on outer ring road. We then studied the improvement in travel time after the introduction of the BPL. We also studied the impact of real time feedback (such as alters for tailgating, over speeding etc.) on driving behaviour. Preparation of a final report of this work is in progress.

March 2020 – July 2020: I worked on the problem of large deviations of the invariant measure in a mean-field model where each particle’s state comes from a countable set (e.g. a system of interacting queues). Such a countable state-space model poses some technical challenges in establishing the LDP. We have made good progress in tacking these technical challenges. As part of COVID-19 response, I worked on building a city-scale agent-based simulator to study the impact of various non-pharmaceutical interventions in Indian cities. This work was done with a larger team of people from IISc and TIFR. We built a state of the art agent-based simulator that captures the interaction among people at various places, and it is capable of implementing a wide variety of interventions such as lockdown, testing and contact tracing, etc. We used the simulator to study the impact of various interventions in Mumbai and Bengaluru. I also worked on (with a larger team from IISc and ISI Bengaluru) the problem of allocating test samples from swab collection centres to COVID-19 testing labs across Karnataka so as to minimise a certain measure of the collective backlog at labs as well as the cost of transporting the samples. Our solution was demonstrated to the Bruhat Bengaluru Mahanagara Palike (BBMP) commissioner.

**Research Publications**

Large deviations of the invariant measure of mean-field interacting particle systems (with R. Sundaresan), in preparation. Large deviations of mean-field interacting particle systems in a fast varying environment (with R. Sundaresan), submitted. Large time behaviour and the second eigenvalue problem for finite state mean-field interacting particle systems (with R. Sundaresan), submitted. City-Scale Agent-Based Simulators for the Study of Non-Pharmaceutical Interventions in the Context of the COVID-19 Epidemic (with IISc-TIFR Agent-Based Simulation Team), submitted to Journal of the Indian Institute of Science. Nonzero-sum Adversarial Hypothesis Testing Games (with P. Loiseau), NeurIPS 2019