Introduction

Machine learning and graph theory are two fields with a lot of synergy and potential for high-level data analysis of system and network structures1. Graph theory, which is the branch of mathematics that deals with graphs as abstract data types that represent relations between objects, is rich in applications to representing data with relational structures such as use in chemistry and molecular biology, social networks and biological systems2. Machine learning techniques especially the graph based learning uses the structure of these graphs for the purpose of making predictions, data classification or pattern recognition3,4. For example, in the process of drug discovery, graphs can present molecules in which atoms are considered nodes and bonds as considered edges; this is used in the venture of determining or predicting chemical properties or biological activities dependent on certain topological features that can be learned by the machine learning algorithms5. In essence, this integration between machine learning and graph theory enhances efficiency in the prediction of elements in areas that depend on the interconnection of components such as computational biology6,7,8, social sciences9 and among others10,11,12. There are many molecular descriptors available but topological indices are most famous and are being widely used in studies due to their ability to find structural features of molecules in a numerical form13.

The topological indexs capacity to discriminate between similar molecules is crucial. Indices provide information about the topology of molecules which is helpful in prediction of physicochemical properties14. Topological indices are mathematical quantities calculated from the molecular graph of a given chemical compound and reflecting structural and connectivity characteristics of the compound. These indices quantify various characteristics of molecular structures such as the shape and size, and atom connectivity without requiring detailed quantum mechanical computations; thus they are effective statistics for investigating chemical properties and biological activities15,16. Some of the most known topological descriptors include the Wiener index which equal to the sum of the distance between all pairs of vertices in the graph and the Zagreb indices which emphasize the Vertex degree17. All these descriptors are extensively employed in Quantitative Structure-Activity Relationships (QSAR) and Quantitative Structure-Property Relationships (QSPR) where the molecular structure is related to solubility, boiling points or biological activity18,19. Topological indices reduce the complicated correlations between molecular characteristics and properties to a single number, which contains the indispensable structural parameters; these descriptive topological indices are particularly essential for predictive modeling in the drug development process, if a machine learning-based approach that can discover patterns and provide accurate predictions of new compounds is involved. Predictive power of QSPR analysis20 has been increased due to recent updates in machine learning models such as Artificial Neural Networks (ANN) and Random Forest (RF) such as antimalaria drugs were studied via QSPR analysis and python based approach for more accurate analysis. Non-linear and complex relationships between topological indices and properties can be handled by these models which is crucial for tasks related to prediction. The correlation between the elements’ functions and characteristics is established by the application of QSPR and QSAR21,22. Within the realms of theoretical and computational chemistry as well as other disciplines, a core idea is the correlation and prediction of physical, chemical, and biological activity based on the structure. Exclusive definitions of the fundamental topological indices are provided for both linked and undirected molecular graphs. Hydrogen atoms are excluded (referred to as “hydrogen suppressed”) and the analysis disregards the inclusion of double bonds and triple bonds. The topological index for a collection of molecular networks may be equal, suggesting that the index is unable to discriminate between the molecular networks.23.

Kulli24 introduced the concept of reverse vertex degree R(v), defined as

$$\begin{aligned} \mathcal {R}(v) = \Delta (G) - d(v) + 1 \end{aligned}$$
(1)

where \(\Delta (G)\) is the maximum degree of the graph G and d(v) is the degree of the vertex v.

Various researchers worked on the QSPR analysis of reverse degree based topological indices for some drugs used in the analysis of para-line graph of Remdesivir25, treatment of H1N126, COVID - 19 drugs27, prediction of physicochemical properties of benzene derivatives28, reverse degree based topological indices and its importance in chemical sciences29.

Various researchers worked on the computation of reverse degree based topological indices for general structures like Dendrimers30, Chain Hex-Derived Network31, hex-derived networks32, Silicate Network33, sodalite materials network34, fullerene cage networks35 and so on. Inspired by the aforesaid ideas, Vignesh et al.36 defined the reduced reverse vertex degree as

$$\begin{aligned} \mathcal{R}\mathcal{R}(v) = \Delta (G) - d(v) + 2 \end{aligned}$$
(2)

where \(\Delta (G)\) is the maximum degree of the graph G and d(v) is the degree of the vertex v.

The reverse degree-based versions of Zagreb indices, harmonic index, forgotten index, shilpa-shanmukha index, atom bond connectivity index, Randic index, sum connectivity index, geometric arithmetic index and the defined indices are dicussed in Table 1, and the reduced reverse degree-based versions of Zagreb indices, harmonic index, forgotten index, shilpa-shanmukha index, atom bond connectivity index, Randic index, sum connectivity index, geometric arithmetic index and the defined indices are dicussed in Table 2. These indices are critical for capturing features that influence physicochemical properties, as demonstrated in prior QSAR/QSPR studies.

In this study, the use of machine learning techniques such as ANN and RF with experimental data makes it easier for researchers to perform elaborate QSPR modeling. These models help to design drugs prospective to enhance their antiviral properties because these molecular models demonstrate how the structure of a molecule influences its biological activity. Therefore, embracing both the structural medicinal chemistry with computational method in relation to the design of other anti-malarial drugs is crucial this is more so with growing outbreaks whereby time is crucial. In conclusion, integrating machine learning to identify new anti-malarial treatment agents shortens the process of identifying better treatment options for the disorder indication that machine learning is a viable solution toward the health concern affecting the entire world. Here we envision to obtain correct predictions of these properties using Reverse and Reduce Reverse topological indices and to implement them in applied machine learning models. Thus, the proposed approach offers not only useful information regarding the molecular properties but also the prospects of applying machine learning to the advancement of drug discovery.

Motivation and significance

This situation is driven by the increasing demand for effective drugs in the global market, particularly for diseases like malaria. This demand puts pressure on the pharmaceutical value chains to discover faster and more efficient methods for producing these drugs. Conventional approaches to drug discovery are usually lengthy and expensive and therefore increased emphasis has to be placed in new technologies. Machine learning has become an invaluable tool in this field, dramatically improving a drug development pipeline through quick identification of the properties and possible effectiveness of a compound. More specifically the application of QSPR analysis in conjunction with Machine Learning yields a framework for approximating the biological activities of antimalarial drugs from their molecular characteristics. With the help of topological descriptors, QSPR models can identify calculations that forecast the pharmacological properties of a compound based on its structure, which will facilitate the identification of better drugs with a faster rate. This approach does not only fasten the identification of more potent antimalarial compounds but also decreases the number of hits that have to go through expensive experiments it therefore represents an important strategy in drug discovery at the present time.

Table 1 Reverse degree-based topological indices.
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Table 2 Reduced reverse degree-based topological indices.
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A bibliometric analysis is shown in Fig. 1 (https://www.vosviewer.com version 1.6.20) clearly demonstrates the interest the world over in the analysis of degree-based topological indicators. This collection of national research investigations improves the field of graph theory through a enhance understanding of the topological indices and a multitude of uses for them. In the keywords bibliometric analysis of the degree-based topological indices in Fig. 2 (https://www.vosviewer.com version 1.6.20), we have presented information in several ways. The study findings show how extensively degree-based topological indices are talked about source (https://www.scopus.com).

Fig. 1
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Bibliometric analysis: different countries degree-based topological indices (https://www.scopus.com).

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Fig. 2
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Bibliometric analysis: topological indices based on degree (https://www.scopus.com).

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Fig. 3
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Antimalaria drugs.

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Material and methods

Fifteen antimalaria drugs were selected for this study. Figure 3a shows their molecular chemical structures of Atovaquone, Artemether and Artemoti and Fig. 3b shows their 2D structures, which include Artelinic acid, Atovaquone, Artemether, Artemotil, Quinine, Artemisinin, Triazole 1, Triazole 2, Triazole 3, Primaquine, Dihydroartemisic acid, Chloroquine, Mefloquine, Halofantrine and Lumefantrine denoted as “A1”, “A2”, “A3”, “A4”, “A5”, “A6”, “A7”, “A8”, “A9”, “A10”, “A11”, “A12”, “A13”, “A14” and “A15”48. To gather the physicochemical properties of the antimalaria drugs, the legacy ChemSpider (https://legacy.chemspider.com) database was utilised. This database is a reliable and lawful source of chemical information. The molecular graphs for each drug are constructed by converting their chemical structures into graph representations using specialised online available software Graph online(https://graphonline.ru/en/ Version V1.0) that records the interactions between atoms and bonds. Important molecular descriptors for the study, the Reverse and Reduced Reverse Topological Indices, were subsequently computed using an algorithm based on Python Programming (www.python.org Version 3.13.2). Using the appropriate Python libraries, two machine learning methods ANN and RF had their feature variables merged with these topological indices. The physicochemical properties of the antimalaria drugs were input into these models during training so that the accuracy and efficiency of each approach could be assessed. The accuracy of the models was evaluated by comparing the predicted values to the actual values using line graphs. This was done by integrating machine learning methodologies with molecular graph data. Tables 3 and 4 represents the values of Reverse topological indices and Reduce Reverse topological indices generated by Algorithm 1 and Algorithm 2, Table 5 represent the physicochemical properties.

Algorithm 1
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Reverse topological indices and molecular descriptor calculation

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Table 3 Actual calculated values for reverse topological indices.
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Algorithm 2
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Reduced reverse topological indices computation

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Table 4 Calculated values for reduce reverse topological indices.
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Table 5 Experimental physiochemical properties of selected drugs.
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Supervised machine learning techniques

In our study two Supervised Machine Learning based models have been employed for prediction. The first included method: ANN and second included method: RF. The general idea is to use these models to predict characteristics that can normally be tested in laboratories while reducing the time and the means that would normally take to do the testing. It is beneficial in handling analysis and computation of such small but large data sets in order to discover chemical behaviors in the shortest possible time, in contrast with the conventional approaches in computations.

Artificial neural network algorithm

Artificial Neural Network is an Machine learning model derived from the structure and function of biological neurons in the human brain. Consisting of interlinked layers of nodes (neurons), these ANN are supposed to make a distinction between given data and give outputs based on examples learnt. The structure of the network which is composed of an input layer; one or more hidden layers and output layer. The neurons personalize information by weighing inputs and applying a kernel activation formula so that the network learns non-linearity in data. This the network achieves by back propagating, a process by which errors in the output are taken as a basis for fine tuning the weights. Due to their learning mechanism, ANN are used for data analysis tasks that involve recognition of patterns such as image analyzers, speech or voice recognition systems and predictive modeling analytical systems. In the area of pharmaceutical science such as drug design, ANN have the capabilities to capture mapping between molecular structure and its corresponding, biological activity or physicochemical property and make prediction in QSPR and QSAR analysis. Due to the efficiency of handling and processing high dimensional data, ANNs are a robust tool for modeling the intricate mapping of input and output variables; hence playing a significant role in enhancing cheminformatics and pharmaceutical based fields. Table 6 represents the predicted values of Reverse indices indices generated by Algorithm-3 and Table 7 represents the error estimates for the predicted values.

Suppose we are given training data \((\textbf{a}_i, \textbf{b}_i)\) where \(\textbf{a}_i \in \mathbb {R}^M\) and \(\textbf{b}_i \in \mathbb {R}^N\). An ANN consists of multiple layers of units, where each unit computes a weighted sum of its inputs followed by an activation function. For the ANN model, we implemented a feedforward neural network using the TensorFlow/Keras library. The network architecture consists of:

  • Input layer The number of neurons is equal to the number of input features.

  • Hidden layers Two fully connected layers with 8 neurons each, using the ReLU activation function.

  • Output layer A single neuron with a linear activation function to predict continuous values.

We optimized the model using the Adam optimizer with a learning rate of \(\alpha = 0.001\). The loss function used was Mean Squared Error (MSE), which minimizes the difference between the predicted and actual values.

The key hyperparameters used in training are:

  • Learning rate: \(0.001\)

  • Batch size: \(32\)

  • Epochs: \(100\)

  • Activation function: ReLU (hidden layers), Linear (output layer)

  • Loss function: Mean Squared Error (MSE)

  • Optimizer: Adam

The pseudocode for training an ANN using backpropagation is as follows and their architecture is presented in Fig. 4 based on reverse and indices respectively:

Algorithm 3
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Artificial neural network (ANN) training

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Fig. 4
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ANN architecture for reverse topological indices.

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Table 6 Predicted values based on reverse indices by using artificial neural networks.
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Table 7 Metrics for reverse and reduce reverse indices ANN.
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Random forest algorithm

Random Forest is a generalize and strong supervise machine learning algorithm, which works by building a large number trees classifier during the training phase then using a majority voting to return a single prediction. With Random Forest each tree in the forest is created by a different random sample and set of features from training data, which helps avoid overfitting. In the case of classification, the output formed is the mean of the votes of all the trees for regression the output formed is the mean of all the responses. Such approach enhances the model stability its accuracy and its ability to perform well on unseen data. In fields like drug discovery the Random Forest is employed for QSPR and QSAR for estimating the biological activity or physicochemical characteristics of chemical compounds depending on structures. RF stands out for its applicability to datasets with numerous predictors and at the same time for the model interpretability; these properties make RF relevant for cheminformatics and pharmaceutical science tasks where the chance of obtaining accurate prediction with a primary focus on potential drug candidates is a priority. Table 8 represents the predicted values of Reverse indices generated by Algorithm 4 and Table 9 represents the error estimates for the predicted values.

Then, the pseudocode of the Random Forest algorithm is as follows:

Algorithm 4
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Random forest regression

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Table 8 Predicted values of reverse topological indices by using random forest.
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Table 9 Metrics for reverse and reduce reverse indices RF.
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Computed vs. actual: physicochemical parameter analysis

In this section, we apply two typical machine learning methods Artificial Neural Networks and Random Forest to evaluate predicted and experimental physicochemical properties. Comparing the results generated by the models with the real data the accuracy of the predicted values of packages of many important parameters was estimated. The comparison was displayed and conducted with the help of a line graph that pointed out the relation between the actual data and the predicted numbers. In the line graph one gets quite clear impression of the performance of models, some of advantages and disadvantages of each kind of methods are shown. While artificial neural networks and random forests were used in different combinations, the accuracy of the qualities prediction varied depending on the case.

Assessing ANN and RF predictions and comparison of error metrics: reverse and reduced reverse indices

Using two different types of topological indices:In the case of the Reverse and Reduced Reverse topological indices we use two machine learning models to predict several physicochemical properties in order to evaluate the performance of both models. These indices are identified as topological descriptors that originate from the architecture of different molecules and are applied to enhance the accuracy of model projection for various chemical and physical events. By applying these models, line graphs display predicted or expected levels of several physicochemical properties such as Boiling Point (Fig. 5), ACD/LogP (Fig. 7), Polarizability (Fig. 9), Enthalpy of Vaporization (Fig. 11). To compare the performance of ANN and RF we employed Mean Squared Error (MSE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and the coefficient of determination (R-Squared) metrics. In all 3D bar graphs we employed (* and **) for Reverse and Reduce Reverse indices respectively. 3D bar graphs that show several error measures used to evaluate model performance are shown in Fig. 6 (Boiling point), Fig. 8 (ACD/LogP), Fig. 10 (Polarizability), Fig. 12 (Enthalpy of Vaporization)and other physiochemical properties displayed in Supplementary file. A portion of each figure indicates the comparison of Four important error metrics; MSE, MAE, RMSE and R-Squared of two different sets of conditions or models differentiated as (*), (**).In addition, the study aims to assess each machine learning models capacity in predicting these physicochemical characteristics together with the topological indices by comparing these predictions. This will provide light on the model’s performance with respect to various chemical descriptors.

Fig. 5
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Comparison for boiling point.

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Fig. 6
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Comparison of boiling point for metrics.

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Fig. 7
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Comparison for ACD/logP.

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Fig. 8
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Comparison of ACD/logP for metrics.

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Fig. 9
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Comparison for polarizability.

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Fig. 10
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Comparison of polarizability for metrics.

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Fig. 11
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Comparison for enthalpy of vaporization.

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Fig. 12
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Comparison of enthalpy of vaporization for metrics.

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Conclusion

In this study two different machine learning algorithms have been employed to predict wide range of physiochemical properties in malaria disease curing drugs. The main objective was to design a good and simple model for prediction that would greatly help the pharmaceutical companies, given that the drug discovery and development is a lengthy process. Using four measures Mean Squared Error , Mean Absolute Error , Root Mean Squared Error and \(R^2\) feasibility of models has been assessed. Among all models, the Reverse ANNs yielded the best results of the broad range of properties that categorized from Table 6 like Boiling Point for Reverse Figs. 5 and 6, ACD/LogP for Reverse Figs. 7 and 8, Polarizability for Reverse Figs. 9 and 10 and Enthalpy of Vaporization for Reverse Figs. 11 and 12 been evaluated. Thus, lowest MSE, MAE and RMSE, in addition to the highest \(R^2\) shown Metrics for Reverse indices of Neural Network with Table 7 depicted it has high predictive reliability and most accurate. On the other hand, Out of all models used in this work, Reduce Reverse ANNs demonstrated especially high performance in predicting the properties, including Molar Refractivity for Reduce Reverse as well as ACD/LogD for Reduce Reverse. The positive signs from preceding glimpses demonstrate its highest capability for which proofs are in the acts of visual comparisons illustrating its predictive accuracy and consistency from one evaluation to another Metrics for Reduce Reverse indices of Neural Network in Table 7. For Density of Reverse , Surface Tension of Reverse, the most accurate prediction was done by using Reverse RF Table 8. In other words, although every one of the algorithms presented has its benefits and pitfalls, ANNs demonstrated the highest level of success for the purpose of employing Reverse and Reduce Reverse topological indices in order to predict the properties of the examined antimalaria drugs. This finding does not only indicate that neural networks can be implemented in pharmaceutical research but also demonstrate that suitable machine learning approaches should be chosen according to the characteristics and needs of chemical datasets used for exploration. Compared with previous studies48, linear regression showed additionally a moderate predictive accuracy, but applying more complex algorithms such as ANN and RF increased the accuracy of the predictions. This shows that out of the three grouping techniques and considering the high non-linearity in the present data, complex models like ANN and RF give improved general predictive accuracy.