  Discriminating between correct and incorrect substrates is a core process in
biology but how is energy apportioned between the conflicting demands of
accuracy ($\mu$), speed ($\sigma$) and total entropy production rate ($P$)?
Previous studies have focussed on biochemical networks with simple structure or
relied on simplifying kinetic assumptions. Here, we use the linear framework
for timescale separation to analytically examine steady-state probabilities
away from thermodynamic equilibrium for networks of arbitrary complexity. We
also introduce a method of scaling parameters that is inspired by Hopfield's
treatment of kinetic proofreading. Scaling allows asymptotic exploration of
high-dimensional parameter spaces. We identify in this way a broad class of
complex networks and scalings for which the quantity $\sigma\ln(\mu)/P$ remains
asymptotically finite whenever accuracy improves from equilibrium, so that
$\mu_{eq}/\mu \to 0$. Scalings exist, however, even for Hopfield's original
network, for which $\sigma\ln(\mu)/P$ is asymptotically infinite, illustrating
the parametric complexity. Outside the asymptotic regime, numerical
calculations suggest that, under more restrictive parametric assumptions,
networks satisfy the bound, $\sigma\ln(\mu/\mu_{eq})/P < 1$, and we discuss the
biological implications for discrimination by ribosomes and DNA polymerase. The
methods introduced here may be more broadly useful for analysing complex
networks that implement other forms of cellular information processing.
