# A Graph-Based k-Nearest Neighbor (KNN) Approach for Predicting Phases in High-Entropy Alloys

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Theoretical Notation Definitions

**Social network graph.**A social network can be mapped to the graph $G\left(V,E\right)$, where $V$ is the node set and $E$ is the edge set.**Neighbors.**A node $u$ is a neighbor of node $v$ in graph $G=\left(V,E\right)$ if there is an edge $\left\{u,v\right\}\in E$.**HEA interaction network.**The HEA compounds are nodes, and the interaction between two compounds are edges that are mapped into a social network [19].**Target compound.**The node considered for phase prediction in the HEA interaction network is called the target compound.**Voting.**The HEA compound is classified by a plurality vote of its neighbors in the KNN algorithm, with the HEA compound being assigned to the phase most common among its k-nearest neighbor.

#### 2.2. Proposed Method

#### 2.3. Evaluation

## 3. Results and Discussion

## 4. Conclusions and Future Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ye, Y.; Wang, Q.; Lu, J.; Liu, C.; Yang, Y. High-entropy alloy: Challenges and prospects. Mater. Today
**2016**, 19, 349–362. [Google Scholar] [CrossRef] - Risal, S.; Zhu, W.; Guillen, P.; Sun, L. Improving phase prediction accuracy for high entropy alloys with machine learning. Comput. Mater. Sci.
**2021**, 192, 110389. [Google Scholar] [CrossRef] - Sheng, G.; Liu, C.T. Phase stability in high entropy alloys: Formation of solid-solution phase or amorphous phase. Prog. Nat. Sci. Mater. Int.
**2011**, 21, 433–446. [Google Scholar] - Chang, Y.-J.; Jui, C.-Y.; Lee, W.-J.; Yeh, A.-C. Prediction of the composition and hardness of high-entropy alloys by machine learning. JOM
**2019**, 71, 3433–3442. [Google Scholar] [CrossRef] - Chanda, B.; Jana, P.P.; Das, J. A tool to predict the evolution of phase and Young’s modulus in high entropy alloys using artificial neural network. Comput. Mater. Sci.
**2021**, 197, 110619. [Google Scholar] [CrossRef] - Dixit, S.; Singhal, V.; Agarwal, A.; Rao, A.P. Multi-label phase-prediction in high-entropy-alloys using Artificial-Neural-Network. Mater. Lett.
**2020**, 268, 127606. [Google Scholar] [CrossRef] - Machaka, R. Machine learning-based prediction of phases in high-entropy alloys. Comput. Mater. Sci.
**2021**, 188, 110244. [Google Scholar] [CrossRef] - Wu, L.; Liu, L.; Wang, Y.; Zhai, Z.; Zhuang, H.; Krishnaraju, D.; Wang, Q.; Jiang, H. A machine learning-based method to design modular metamaterials. Extrem. Mech. Lett.
**2020**, 36, 100657. [Google Scholar] [CrossRef] - Agarwal, A.; Prasada Rao, A. Artificial intelligence predicts body-centered-cubic and face-centered-cubic phases in high-entropy alloys. JOM
**2019**, 71, 3424–3432. [Google Scholar] [CrossRef] - Krishna, Y.V.; Jaiswal, U.K.; Rahul, M. Machine learning approach to predict new multiphase high entropy alloys. Scr. Mater.
**2021**, 197, 113804. [Google Scholar] [CrossRef] - Jalali, M.; Tsotsalas, M.; Wöll, C. MOFSocialNet: Exploiting Metal-Organic Framework Relationships via Social Network Analysis. Nanomaterials
**2022**, 12, 704. [Google Scholar] [CrossRef] [PubMed] - Rickman, J.; Balasubramanian, G.; Marvel, C.; Chan, H.; Burton, M.-T. Machine learning strategies for high-entropy alloys. J. Appl. Phys.
**2020**, 128, 221101. [Google Scholar] [CrossRef] - Bhandari, U.; Rafi, M.R.; Zhang, C.; Yang, S. Yield strength prediction of high-entropy alloys using machine learning. Mater. Today Commun.
**2021**, 26, 101871. [Google Scholar] [CrossRef] - Wen, C.; Wang, C.; Zhang, Y.; Antonov, S.; Xue, D.; Lookman, T.; Su, Y. Modeling solid solution strengthening in high entropy alloys using machine learning. Acta Mater.
**2021**, 212, 116917. [Google Scholar] [CrossRef] - Qiao, L.; Liu, Y.; Zhu, J. A focused review on machine learning aided high-throughput methods in high entropy alloy. J. Alloys Compd.
**2021**, 877, 160295. [Google Scholar] [CrossRef] - Lee, S.Y.; Byeon, S.; Kim, H.S.; Jin, H.; Lee, S. Deep learning-based phase prediction of high-entropy alloys: Optimization, generation, and explanation. Mater. Des.
**2021**, 197, 109260. [Google Scholar] [CrossRef] - Yan, Y.; Lu, D.; Wang, K. Accelerated discovery of single-phase refractory high entropy alloys assisted by machine learning. Comput. Mater. Sci.
**2021**, 199, 110723. [Google Scholar] [CrossRef] - Jaiswal, U.K.; Krishna, Y.V.; Rahul, M.; Phanikumar, G. Machine learning-enabled identification of new medium to high entropy alloys with solid solution phases. Comput. Mater. Sci.
**2021**, 197, 110623. [Google Scholar] [CrossRef] - Ghouchan Nezhad Noor Nia, R.; Jalali, M.; Mail, M.; Ivanisenko, Y.; Kübel, C. Machine Learning Approach to Community Detection in a High-Entropy Alloy Interaction Network. ACS Omega
**2022**, 7, 12978–12992. [Google Scholar] [CrossRef] - Visa, S.; Ramsay, B.; Ralescu, A.L.; Van Der Knaap, E. Confusion matrix-based feature selection. MAICS
**2011**, 710, 120–127. [Google Scholar] - Armah, G.K.; Luo, G.; Qin, K. A deep analysis of the precision formula for imbalanced class distribution. Int. J. Mach. Learn. Comput.
**2014**, 4, 417–422. [Google Scholar] [CrossRef] [Green Version] - Berrar, D. Cross-Validation. In Reference Module in Life Sciences; Elsevier: Amsterdam, The Netherlands, 2019. [Google Scholar]

**Figure 1.**An example of the proposed method for phase prediction in HEAs. In Step (

**a**), we extracted the related features/descriptors from the HEA database, and then (

**b**) created an interaction network based on the similarities between HEAs. (

**c**) In this examplem ZrHfTiCuNi alloy is considered as a phase prediction sample, hence related community of this HEAs is extracted from the interaction network. (

**d**) The ZrHfTiCuNi has four neighbors; therefore, three of them were selected which are highlighted in dark blue if k is three. (

**e**) Finally, phase can be predicted by voting on neighbors’ labels as Amorphous for ZrHfTiCuNi.

**Figure 2.**A portion of the HEA interaction network with Fruchterman Reingold layout. Nodes with a larger circle have more neighbors, which indicates a stronger relationship.

**Figure 3.**The probability distribution of the degree of a node in an HEA network is the distribution of connections it has to other nodes which is called degree distribution.

**Figure 4.**Part of the HEA interaction network with compounds neighbors. The ZrHFTiCuNi alloy is a sample with three of the most similar neighbors, as shown in dark blue.

**Figure 5.**The comparison of the proposed approach comparison with other models; its prediction is more precise.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ghouchan Nezhad Noor Nia, R.; Jalali, M.; Houshmand, M.
A Graph-Based k-Nearest Neighbor (KNN) Approach for Predicting Phases in High-Entropy Alloys. *Appl. Sci.* **2022**, *12*, 8021.
https://doi.org/10.3390/app12168021

**AMA Style**

Ghouchan Nezhad Noor Nia R, Jalali M, Houshmand M.
A Graph-Based k-Nearest Neighbor (KNN) Approach for Predicting Phases in High-Entropy Alloys. *Applied Sciences*. 2022; 12(16):8021.
https://doi.org/10.3390/app12168021

**Chicago/Turabian Style**

Ghouchan Nezhad Noor Nia, Raheleh, Mehrdad Jalali, and Mahboobeh Houshmand.
2022. "A Graph-Based k-Nearest Neighbor (KNN) Approach for Predicting Phases in High-Entropy Alloys" *Applied Sciences* 12, no. 16: 8021.
https://doi.org/10.3390/app12168021