## 3 Results  
We implemented our method of Steps 1 to 5 for inferring rank-2 chemical graphs and conducted experiments to evaluate the computational efficiency for three chemical properties \(\pi\): octanol/water partition coefficient (\(\mathsf{K}_{\infty}\)), melting point (\(\mathsf{M}\textsc{p}\)), and boiling point (\(\mathsf{B}\textsc{p}\)). We executed the experiments on a PC with Intel Core i5 1.6 GHz CPU and 8 GB of RAM running under the Mac OS operating system version 10.14.6. We show 2D drawings of some of the inferred chemical graphs, where ChemDoodle version 10.2.0 is used for constructing the drawings. **Results on Phase 1. **  
**Step 1. **: We set a graph class \(\mathcal{G}\) to be the set of all rank-2 chemical graphs. For each property \(\pi\in\{\mathsf{K}_{\infty}\), \(\mathsf{M}\textsc{p}\), \(\mathsf{B}\textsc{p}\}\), we select a set \(\Lambda\) of chemical elements and collected a data set \(D_{\pi}\) on rank-2 chemical graphs over \(\Lambda\) provided by HSDB from PubChem. To construct the data set, we eliminated chemical compounds that have at most three carbon atoms or contain a charged element such as \(\texttt{N}^{+}\) or an element a \(\in\Lambda\) in which the valence is different from our setting of valence function val. Table 1 shows the size and range of data sets that we prepared for each chemical property in Step 1, where we denote the following:  
* \(\pi\): one of the chemical properties \(\mathsf{K}_{\text{cov}}\), \(\mathsf{M}\)p and \(\mathsf{B}\)p;
* \(|D_{\pi}|\): the size of data set \(D_{\pi}\) for property \(\pi\);
* \(\Lambda\): the set of chemical elements over data set \(D_{\pi}\) (hydrogen atoms are added at the final stage);
* \(|\Gamma|\): the number of tuples in \(\Gamma\);
* \([\underline{u},\overline{n}]\): the minimum and maximum number \(n(G)\) of non-hydrogen atoms over data set \(D_{\pi}\);
* \([\underline{\mathsf{cs}},\overline{\mathsf{cs}}]\), \([\underline{\mathsf{ch}},\overline{\mathsf{ch}}]\): the minimum and maximum core size and core height over chemical compounds in \(D_{\pi}\), respectively;
* \([\underline{\theta},\overline{\theta}]\): the minimum and maximum values of the topological parameter \(\theta(G)\) over data set \(D_{\pi}\) and
* \([\underline{a},\overline{a}]\): the minimum and maximum values of \(a(G)\) in \(\pi\) over data set \(D_{\pi}\). Step 2 We used a feature function \(f\) that consists of the descriptors defined in Section 2.1. Step 3 We used scikit-learn version 0.21.6 with Python 3.7.4 to construct ANNs \(\mathcal{N}\) where the tool and activation function are set to be MLPRegressor and ReLU, respectively. We tested several different architectures of ANNs for each chemical property. To evaluate the performance of the resulting prediction function \(\psi_{N}\) with cross-validation, we partition a given data set \(D_{\pi}\) into five subsets \(D_{\pi}^{(i)}\), \(i\in[1,5]\) randomly, where \(D_{\pi}\setminus D_{\pi}^{(i)}\) is used for a training set and \(D_{\pi}^{(i)}\) is used for a test set in five trials \(i\in[1,5]\). For a set \(\{y_{1},y_{2},\ldots,y_{N}\}\) of observed values and a set \(\{\psi_{1},\psi_{2},\ldots,\psi_{N}\}\) of predicted values, we define the coefficient of determination to be \(\mathrm{R}^{2}\triangleq 1-\frac{\sum_{j\in[1,N]}(y_{j}-\overline{y})^{2}}{ \sum_{j\in[1,N]}(y_{j}-\overline{y})^{2}}\), where \(\overline{y}=\frac{1}{N}\sum_{j\in[1,N]}y_{j}\). Table 2 shows the results on Steps 2 and 3, where  
* \(k\): the number of descriptors for the chemical compounds in data set \(D_{\pi}\) for property \(\pi\);
* Activation: the choice of activation function;
* Architecture: \((a,b,1)\) consists of an input layer with \(a\) nodes, a hidden layer with \(b\) nodes, and an output layer with a single node, where \(a\) is equal to the number of descriptors;
* L-time: the average time (sec.)