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Nonequilibrium reaction networks (NRNs) underlie most biological functions.Despite their diverse dynamic properties, NRNs share the signaturecharacteristics of persistent probability fluxes and continuous energydissipation even in the steady state. Dynamics of NRNs can be described atdifferent coarse-grained levels. Our previous work showed that the apparentenergy dissipation rate at a coarse-grained level follows an inverse power lawdependence on the scale of coarse-graining. The scaling exponent is determinedby the network structure and correlation of stationary probability fluxes.However, it remains unclear whether and how the (renormalized) flux correlationvaries with coarse-graining. Following Kadanoff's real space renormalizationgroup (RG) approach for critical phenomena, we address this question bydeveloping a State-Space Renormalization Group (SSRG) theory for NRNs, whichleads to an iterative RG equation for the flux correlation function. In squareand hypercubic lattices, we solve the RG equation exactly and find two types offixed point solutions: a family of nontrivial fixed points where thecorrelation exhibits power-law decay and a trivial fixed point where thecorrelation vanishes beyond the nearest neighbors. The power-law fixed point isstable if and only if the power exponent is less than the lattice dimension$n$. Consequently, the correlation function converges to the power-law fixedpoint only when the correlation in the fine-grained network decays slower than$r^{-n}$ and to the trivial fixed point otherwise. If the flux correlation inthe fine-grained network contains multiple stable solutions with differentexponents, the RG iteration dynamics select the fixed point solution with thesmallest exponent. We also discuss a possible connection between the RG flowsof flux correlation with those of the Kosterlitz-Thouless transition.

State-space renormalization group theory of nonequilibrium reaction networks: Exact solutions for hypercubic lattices in arbitrary dimensions