Algorithm 1 Constructing the underlying space of a fitness landscape of GRNs

Require: The fixed underlying collections of loci $\mathsf{\Gamma }$ and proteins $\mathsf{\Omega }$ of GRNs Ensure: The representative GRN $g_{\theta }$ of each equivalence class $\theta \in \mathsf{\Theta }$, and its number of mutational neighbors $A_{g_{\theta }}\left({\theta }^{\prime }\right)$ in any equivalence class ${\theta }^{\prime }\in \mathsf{\Theta }$ 1:$k\leftarrow 0$                            ▹ initialization 2:$g_{{\theta }_0}\leftarrow$ a GRN with no self-loop, where ${\theta }_0$ is the only equivalence class in layer ${\mathsf{\Theta }}_0$ 3:Store the phenotype-preserving automorphisms ${\mathsf{\Sigma }}_x\left(g_{{\theta }_0}\right)$. 4:Compute $A_{g_{{\theta }_0}}\left({\theta }_0\right)$ via Equation (5). 5:while$k<\left|\mathsf{\Gamma }\right|$do                    ▹ incrementally find $\mathsf{\Theta }$ 6:    Construct and store the joining operations $\mathsf{\Phi }$ for layer ${\mathsf{\Theta }}_k$. 7:    ${\mathcal{M}}_{\mathsf{\Phi }}^{+}\leftarrow$ grouping of ${\mathcal{M}}^{+}\left({\mathsf{\Theta }}_k\right)$ acted by $\mathsf{\Phi }$ 8:    ${\mathsf{\Theta }}_{k+1}$ corresponds to the groups in ${\mathcal{M}}_{\mathsf{\Phi }}^{+}$. 9:    for all ${\theta }^{\prime }\in {\mathsf{\Theta }}_{k+1}$ do 10:        $g_{{\theta }^{\prime }}\leftarrow$ a GRN in ${\mathcal{M}}_{\mathsf{\Phi }}^{+}\left({\theta }^{\prime }\right)$          ▹ choose the representative GRN 11:        Store the phenotype-preserving automorphisms ${\mathsf{\Sigma }}_x\left(g_{{\theta }^{\prime }}\right)$. 12:    end for 13:    for all $\theta \in {\mathsf{\Theta }}_k,{\theta }^{\prime }\in {\mathsf{\Theta }}_{k+1}$do     ▹ count the number of mutational neighbors 14:        Compute $A_{g_{\theta }}\left({\theta }^{\prime }\right)$ and $A_{g_{{\theta }^{\prime }}}\left(\theta \right)$ via Equations (1) and (3). 15:    end for 16:    for all ${\theta }_1^{\prime },{\theta }_2^{\prime }\in {\mathsf{\Theta }}_{k+1}$ do 17:        Compute $A_{g_{{\theta }_1^{\prime }}}\left({\theta }_2^{\prime }\right)$ via Equations (4) and (5). 18:    end for 19:    $k\leftarrow k+1$ 20:end while 21:Set any remaining, not computed $A_{g_{\theta }}\left({\theta }^{\prime }\right)$ to zero.