Scalable tests for ergodicity analysis of large-scale interconnected stochastic reaction networks Corentin Briat, Ankit Gupta, Iman Shames and Mustafa Khammash Index Terms— Stochastic reaction networks; Markov pro- cesses; reaction network theory; ergodicity; systems biology I. I NTRODUCTION A wide variety of processes arising in chemistry, bio- chemistry, ecology, epidemiology and social sciences can be represented as reaction networks [1]–[3]. The core idea is to represent a given system in terms of the evolution of a population of different agents. The time-evolution of the re- spective populations is described by a set of reactions, which, when they take place, change the populations in a fixed and given manner. Deterministic reaction networks, repre- sented in terms of ordinary, functional or partial differential equations, have been historically considered first. Famous examples are the Lotka-Volterra equations in ecology and SIR-models in epidemiology; see e.g. [4]. A deterministic representation of networks remains valid as long as the pop- ulations of the agents are large. However, when populations are small, random effects become dominant and cannot be neglected anymore. Under a well-mixed assumption, it has been shown that the evolution of the populations can be described by a Markov process [5]. The case of low-copy numbers arises quite frequently in biochemistry of the cell [6] where randomness can lead to important discrepancies in the behavior of identical organisms. More surprisingly, certain biological circuits have been shown to exploit randomness in order to achieve their function in an efficient way; see e.g. [7]–[9]. The behavior of deterministic models can be analyzed using the rich underlying mathematical theory of determin- istic dynamical systems and differential equations. Stability properties can be, for example, studied using Lyapunov theory. When it comes to stochastic reaction networks, the analysis becomes much more complicated due to the random nature of the system. Fewer tools moreover available for stochastic reaction networks; see e.g. [10]–[12]. The goal of this paper/talk, is to provide an efficient way for analyzing the long-term behavior of stochastic reaction networks. We first recall some conditions for es- tablishing (structural) ergodicity and moments bounded- Corentin Briat, Ankit Gupta and Mustafa Khammash are with the Department of Biosystems Science and Engineering (D- BSSE), Swiss Federal Institute of Technology–Z¨ urich (ETH-Z), Basel, Switzerland; email: corentin.briat@bsse.ethz.ch, ankit.gupta@bsse.ethz.ch, mustafa.khammash@bsse. ethz.ch, url: http://www.silva.bsse.ethz.ch/ctsb/, http://www.briat.info. Iman Shames is with the Department of Electrical and Electronic Engineering, University of Melbourne, Melbourne, Australia; email: iman.shames@unimelb.edu.au. ness/convergence of a given reaction network that have been developed in [12]. Ergodicity, for stochastic processes, is analogous to having a globally asymptotically stable equilibrium point for deterministic dynamics [13]. It thus provides a suitable stability notion in our context since it guarantees that the stochastic reaction network has a unique attractive stationary distribution. We then extend these results to address the problem of establishing ergodicity of intercon- nected reaction networks, that is, ergodicity of a network of networks. We show that ergodicity of interconnected networks can be expressed through local (to each network) ergodicity criteria taking the form of linear programs that involve additional parameters. It is emphasized that the scalability of the approach can be enhanced by relying on a distributed implementation of the dual optimization problem. Several examples are given for illustration. Notations: The set of whole numbers is denoted by N 0 , the set of positive real numbers by R >0 and the set of integers by Z. For x, y ∈ R n , 〈x, y〉 is the standard inner-product on R n . The set of symmetric matrices of dimension n is denoted by S n . The vector column made of the terms x 1 ,...,x n is denoted by col i (x i ). II. STOCHASTIC REACTION NETWORKS Let us consider a reaction network with d species S 1 ,..., S d and K reactions R 1 ,...,R K . For each reaction R k , we associate the stoichiometric vector ζ k ∈ Z d which describes how the populations change when the reaction R k fires. That is, if the state is x and the reaction R k fires, then the state immediately after the reaction is given by x+ζ k . The propensity function of reaction R k is denoted by λ k (x) ≥ 0 (see Table I) which means that, when the state is equal to x, the reaction R k fires after a random time that is exponentially distributed with rate λ k (x). TABLE I LIST OF THE CONSIDERED REACTIONS AND THEIR PROPENSITY FUNCTION Type Reaction Propensity λ(x) 1 ∅ k −−−→ S 1 k 2 S i k −−−→ · kx i 3 S i + S i k −−−→ · k 2 x i (x i − 1) 4 S i + S j k −−−→ · kx i x j The evolution of the system is represented as a Markov process (X x0 (t)) t≥0 where x 0 is the initial state. The popula- 21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. Groningen, The Netherlands ISBN: 978-90-367-6321-9 92