Graphical symmetries of reaction networks and their stationary measures
Reaction networks are used to model chemical reactions, biochemical reactions, interacting populations in ecology, genetics and infectious disease dynamics among many other systems. Depending on the system scale, three most commonly used models are continuous-time Markov chain (CTMC), stochastic differential equations and ordinary differential equations (ODE). With increased data and scientific focus shifting towards biochemistry which often involves populations as small as few tens of molecules, CTMC models have seen a dramatic increase in relevance.
The stationary distribution of a continuous-time Markov chain model describes its long-term dynamics. In general, it is impossible to obtain the explicit form of the stationary distribution, even for very small networks. However, when the network has some natural symmetries, we can obtain the explicit stationary distribution, and it can be connected with the equilibria of the corresponding ODE system. Prof. Joshi will discuss the symmetries of reaction balance, complex balance, reaction vector balance, and cycle balance and their properties in both the deterministic setting (ODE) and stochastic setting (CTMC). He will also discuss the relations between the various forms of balance and the connection with existence and form of the stationary distribution.