Clustering-based ℌ2-state aggregation of positive networks and its application to reduction of chemical master equations

Abstract

In this paper, based on a notion of network clustering, we propose a state aggregation method for positive systems evolving over directed networks, which we call positive networks. In the proposed method, we construct a set of clusters (i.e., disjoint sets of state variables) according to a kind of local uncontrollability of systems. This method preserves interconnection topology among clusters as well as stability and some particular properties, such as system positivity and steady-state characteristic (steady-state distribution). In addition, we derive an ℌ2-error bound of the state discrepancy caused by the aggregation. The efficiency of the proposed method is shown through the reduction of a chemical master equation representing the time evolution of the Michaelis-Menten chemical reaction system.