Algorithm 2 graph grammar based optimization of graph structure and embedment

(1)Initialize the current graph embedment by some graph topology and some initial map {G0, φ0(G0)}. (2)For the current graph embedment {Gi, φi(Gi)}, apply all grammar operations from a grammar set {Ψ1,…, Ψr}, and generate a set of s candidate graph injections $\{D_{}^k,\varphi \left(D_{}^k\right), k=1\dots s\}.$ (3)Further optimize each candidate graph embedment using Algorithm 1, and obtain a set of s energy values $\left\{{\overline{U}}^{\varphi \left(D_{}^k\right)}\left(D_{}^k\right)\right\}$ (4)Among all candidate graph injections, select an injection map with the minimum energy as $\left\{G_{i+1}^{},{\varphi }_{i+1}^{}(G_{i+1}^{})\right\}.$ Repeat 2–4 until the graph contains a required number of nodes.