We study the *convergence time* (time to reach a stable state) of switching networks; in each column, one of two stable states can be reached from the initial conditions. All the reaction rates are set equal and all the species start at equal quantities (details are provided in Supplementary Materials). Columns A-D concern the DC, AM, SC, CC switches respectively. Row 1 gives a depiction of the networks from which one can precisely recover the chemical reaction networks according to our notation. A *catalytic reaction* is represented by a circle on top of an arrow; for example, the top right of (F) contains the reaction *b*+*z*→*z*+*y* with catalyst *z*. A reaction like *x*+*y*→*y*+*y* is said to be *autocatalytic*. The full diagram (F) depicts two catalysts, *z* and *r*, acting on species *x* and *y* through a shared intermediary *b*, representing the 4 reactions *x*+*z*→*z*+*b*, *b*+*z*→*z*+*y*, *y*+*r*→*r*+*b*, and *b*+*r*→*r*+*x*. We use a more compact pinched-arrow graphical notation (E) to represent the same network as (F), hiding the intermediary species *b* that is assumed not to enter any other reaction. Note that if multiple catalysts act on the same pinched arrow (as in Fig. 2), they all act on the hidden intermediary in the same way. Row 2 contains deterministic simulations for the mass action ODEs of the respective systems for four values of the initial discrepancy between *x* and *y*, from 10% to 0.01% (not meaningful for DC as the system would rest at its initial setting), (Y-axis scale on the right). Row 3 shows sample stochastic simulations consisting of individual traces (black lines) of the Gillespie algorithm. The background heat maps, shown in logarithmic scale, give the probability *Pr*(*x*_{i}|t_{j}) that a system will have *x*_{i} molecules of *x* at time *t*_{j}. The horizontal axis is time, and the vertical axis is concentration (row 2) or number of molecules (row 3). Subscripts ‘s’ are for stochastic variables and ‘p’ for probabilistic variables.

## PubMed Commons