Research article Special Issues

Modeling and diagnosis Parkinson disease by using hand drawing: deep learning model

  • Received: 09 December 2023 Revised: 19 January 2024 Accepted: 29 January 2024 Published: 19 February 2024
  • MSC : 11Y40, 11Y16, 68T10, 68Q32

  • Patients with Parkinson's disease (PD) often manifest motor dysfunction symptoms, including tremors and stiffness. The presence of these symptoms may significantly impact the handwriting and sketching abilities of individuals during the initial phases of the condition. Currently, the diagnosis of PD depends on several clinical investigations conducted inside a hospital setting. One potential approach for facilitating the early identification of PD within home settings involves the use of hand-written drawings inside an automated PD detection system for recognition purposes. In this study, the PD Spiral Drawings public dataset was used for the investigation and diagnosis of PD. The experiments were conducted alongside a comparative analysis using 204 spiral and wave PD drawings. This study contributes by conducting deep learning models, namely DenseNet201 and VGG16, to detect PD. The empirical findings indicate that the DenseNet201 model attained a classification accuracy of 94% when trained on spiral drawing images. Moreover, the model exhibited a receiver operating characteristic (ROC) value of 99%. When comparing the performance of the VGG16 model, it was observed that it attained a better accuracy of 90% and exhibited a ROC value of 98% when trained on wave images. The comparative findings indicate that the outcomes of the proposed PD system are superior to existing PD systems using the same dataset. The proposed system is a very promising technological approach that has the potential to aid physicians in delivering objective and dependable diagnoses of diseases. This is achieved by leveraging important and distinctive characteristics extracted from spiral and wave drawings associated with PD.

    Citation: Theyazn H. H. Aldhyani, Abdullah H. Al-Nefaie, Deepika Koundal. Modeling and diagnosis Parkinson disease by using hand drawing: deep learning model[J]. AIMS Mathematics, 2024, 9(3): 6850-6877. doi: 10.3934/math.2024334

    Related Papers:

  • Patients with Parkinson's disease (PD) often manifest motor dysfunction symptoms, including tremors and stiffness. The presence of these symptoms may significantly impact the handwriting and sketching abilities of individuals during the initial phases of the condition. Currently, the diagnosis of PD depends on several clinical investigations conducted inside a hospital setting. One potential approach for facilitating the early identification of PD within home settings involves the use of hand-written drawings inside an automated PD detection system for recognition purposes. In this study, the PD Spiral Drawings public dataset was used for the investigation and diagnosis of PD. The experiments were conducted alongside a comparative analysis using 204 spiral and wave PD drawings. This study contributes by conducting deep learning models, namely DenseNet201 and VGG16, to detect PD. The empirical findings indicate that the DenseNet201 model attained a classification accuracy of 94% when trained on spiral drawing images. Moreover, the model exhibited a receiver operating characteristic (ROC) value of 99%. When comparing the performance of the VGG16 model, it was observed that it attained a better accuracy of 90% and exhibited a ROC value of 98% when trained on wave images. The comparative findings indicate that the outcomes of the proposed PD system are superior to existing PD systems using the same dataset. The proposed system is a very promising technological approach that has the potential to aid physicians in delivering objective and dependable diagnoses of diseases. This is achieved by leveraging important and distinctive characteristics extracted from spiral and wave drawings associated with PD.



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    [1] C. G. Goetz, The history of Parkinson's disease: early clinical descriptions and neurological therapies, Cold Spring Harb. Perspect. Med., 1 (2011), a008862. https://doi.org/10.1101/cshperspect.a008862 doi: 10.1101/cshperspect.a008862
    [2] S. D. Vassar, Y. M. Bordelon, R. D. Hays, N. Diaz, R. Rausch, C. Mao, et al., Confirmatory factor analysis of the motor unified Parkinson's disease rating scale, Park. Dis., 2012 (2012), 719167. https://doi.org/10.1155/2012/719167 doi: 10.1155/2012/719167
    [3] Parkinson disease, World Health Organization, 9 August 2023. Available online: https://www.who.int/news-room/fact-sheets/detail/parkinson-disease.
    [4] M. C. De Rijk, L. J. Launer, K. Berger, M. M. Breteler, J. F. Dartigues, M. Baldereschi, et al., Prevalence of Parkinson's disease in Europe: a collaborative study of population-based cohorts: neurologic diseases in the elderly research group, Neurology, 54 (2000), S21–S23.
    [5] İ. Cantürk, F. Karabiber, A machine learning system for the diagnosis of Parkinson's disease from speech signals and its application to multiple speech signal types, Arab. J. Sci. Eng., 41 (2016), 5049–5059. https://doi.org/10.1007/s13369-016-2206-3 doi: 10.1007/s13369-016-2206-3
    [6] N. Singh, V. Pillay, Y. E. Choonara, Advances in the treatment of Parkinson's disease, Prog. Neurobiol., 81 (2007), 29–44. https://doi.org/10.1016/j.pneurobio.2006.11.009 doi: 10.1016/j.pneurobio.2006.11.009
    [7] A. Rana, A. S. Rawat, A. Bijalwan, H. Bahuguna, Application of multi-layer (perceptron) artificial neural network in the diagnosis system: a systematic review, 2018 International Conference on Research in Intelligent and Computing in Engineering (RICE), IEEE, 2018, 1–6. https://doi.org/10.1109/RICE.2018.8509069
    [8] L. F. Gomez-Gomez, A. Morales, J. Fierrez, J. R. Orozco-Arroyave, Exploring facial expressions and affective domains for Parkinson detection, arXiv, 2020. https://doi.org/10.48550/arXiv.2012.06563
    [9] A. M. García, T. Arias-Vergara, J. C. Vasquez-Correa, E. Nöth, M. Schuster, A. E. Welch, et al., Cognitive determinants of dysarthria in Parkinson's disease: an automated machine learning approach, Mov. Disord., 36 (2021), 2862–2873. https://doi.org/10.1002/mds.28751 doi: 10.1002/mds.28751
    [10] A. Birba, I. García-Cordero, G. Kozono, A. Legaz, A. Ibáñez, L. Sedeño, et al., Losing ground: frontostriatal atrophy disrupts language embodiment in Parkinson's and Huntington's disease, Neurosci. Biobehav. Rev., 80 (2017), 673–687. https://doi.org/10.1016/j.neubiorev.2017.07.011 doi: 10.1016/j.neubiorev.2017.07.011
    [11] J. Dolz, C. Desrosiers, I. B. Ayed, 3D fully convolutional networks for subcortical segmentation in MRI: a large-scale study, NeuroImage, 170 (2018), 456–470. https://doi.org/10.1016/j.neuroimage.2017.04.039 doi: 10.1016/j.neuroimage.2017.04.039
    [12] M. Ghafoorian, N. Karssemeijer, T. Heskes, I. W. van Uden, C. I. Sanchez, G. Litjens, et al., Location sensitive deep convolutional neural networks for segmentation of white matter hyperintensities, Sci. Rep., 7 (2017), 5110. https://doi.org/10.1038/s41598-017-05300-5 doi: 10.1038/s41598-017-05300-5
    [13] S. H. Wang, P. Phillips, Y. Sui, B. Liu, M. Yang, H. Cheng, Classification of Alzheimer's disease based on eight-layer convolutional neural network with leaky rectified linear unit and max pooling, J. Med. Syst., 42 (2018), 85. https://doi.org/10.1007/s10916-018-0932-7 doi: 10.1007/s10916-018-0932-7
    [14] M. Younis Thanoun, M. T. Yaseen, A comparative study of Parkinson disease diagnosis in machine learning, Proceedings of the 2020 the 4th International Conference on Advances in Artificial Intelligence, 2020, 23–28. https://doi.org/10.1145/3441417.3441425 doi: 10.1145/3441417.3441425
    [15] T. Elhassan, M. Aljurf, Classification of imbalance data using Tomek link (T-link) combined with random under-sampling (RUS) as a data reduction method, Glob. J. Technol. Optim., 1 (2016), 1–11. https://doi.org/10.4172/2229-8711.S1:111 doi: 10.4172/2229-8711.S1:111
    [16] S. Fan, Y. Sun, Early detection of Parkinson's disease using machine learning and convolutional neural networks from drawing movements, CS & IT Conference Proceedings, 12 (2022), 291–301. https://doi.org/10.5121/csit.2022.121523 doi: 10.5121/csit.2022.121523
    [17] P. Arora, A. Mishra, A. Malhi, Machine learning Ensemble for the Parkinson's disease using protein sequences, Multimed. Tools Appl., 81 (2022), 32215–32242. https://doi.org/10.1007/s11042-022-12960-7 doi: 10.1007/s11042-022-12960-7
    [18] Shivangi, A. Johri, A. Tripathi, Parkinson disease detection using deep neural networks, 2019 Twelfth International Conference on Contemporary Computing (IC3), 2019, 1–4. https://doi.org/10.1109/IC3.2019.8844941 doi: 10.1109/IC3.2019.8844941
    [19] D. A. Rastegar, N. Ho, G. M. Halliday, N. Dzamko, Parkinson's progression prediction using machine learning and serum cytokines, npj Parkinsons. Dis., 5 (2019), 14. https://doi.org/10.1038/s41531-019-0086-4 doi: 10.1038/s41531-019-0086-4
    [20] M. Nilashi, H. Ahmadi, A. Sheikhtaheri, R. Naemi, R. Alotaibi, A. A. Alarood, et al., Remote tracking of Parkinson's disease progression using ensembles of deep belief network and self-organizing map, Expert Syst. Appl., 159 (2020), 113562. https://doi.org/10.1016/j.eswa.2020.113562 doi: 10.1016/j.eswa.2020.113562
    [21] R. Das, A comparison of multiple classification methods for diagnosis of Parkinson disease, Expert Syst. Appl., 37 (2010), 1568–1572. https://doi.org/10.1016/j.eswa.2009.06.040 doi: 10.1016/j.eswa.2009.06.040
    [22] Saloni, R. K. Sharma, A. K. Gupta, Voice analysis for telediagnosis of Parkinson disease using artificial neural networks and support vector machines, Int. J. Intell. Syst. Appl., 7 (2015), 41–47. https://doi.org/10.5815/ijisa.2015.06.04 doi: 10.5815/ijisa.2015.06.04
    [23] H. L. Chen, G. Wang, C. Ma, Z. N. Cai, W. B. Liu, S. J. Wang, An efficient hybrid kernel extreme learning machine approach for early diagnosis of Parkinson's disease, Neurocomputing, 184 (2016), 131–144. https://doi.org/10.1016/j.neucom.2015.07.138 doi: 10.1016/j.neucom.2015.07.138
    [24] N. M. Tahir, Parkinson disease gait classification based on machine learning approach, J. Appl. Sci., 12 (2012), 180–185. https://doi.org/10.3923/jas.2012.180.185 doi: 10.3923/jas.2012.180.185
    [25] E. Abdulhay, N. Arunkumar, K. Narasimhan, E. Vellaiappan, V. Venkatraman, Gait and tremor investigation using machine learning techniques for the diagnosis of Parkinson disease, Futur. Gener. Comput. Syst., 83 (2018), 366–373. https://doi.org/10.1016/j.future.2018.02.009 doi: 10.1016/j.future.2018.02.009
    [26] Y. N. Zhang, Can a smartphone diagnose Parkinson disease? A deep neural network method and telediagnosis system implementation, Park. Dis., 2017 (2017), 6209703. https://doi.org/10.1155/2017/6209703 doi: 10.1155/2017/6209703
    [27] G. Nagasubramanian, M. Sankayya, Multi-Variate vocal data analysis for detection of Parkinson disease using deep learning, Neural Comput. Appl., 33 (2020), 4849–4864. https://doi.org/10.1007/s00521-020-05233-7 doi: 10.1007/s00521-020-05233-7
    [28] I. El Maachi, G. A. Bilodeau, W. Bouachir, Deep 1D-convnet for accurate Parkinson disease detection and severity prediction from gait, Expert Syst. Appl., 143 (2020), 113075. https://doi.org/10.1016/j.eswa.2019.113075 doi: 10.1016/j.eswa.2019.113075
    [29] A. Naseer, M. Rani, S. Naz, M. I. Razzak, M. Imran, G. Xu, Refining Parkinson's neurological disorder identification through deep transfer learning, Neural Comput. Appl., 32 (2020), 839–854. https://doi.org/10.1007/s00521-019-04069-0 doi: 10.1007/s00521-019-04069-0
    [30] S. Shinde, S. Prasad, Y. Saboo, R. Kaushick, J. Saini, P. K. Pal, et al., Predictive markers for Parkinson's disease using deep neural nets on neuromelanin sensitive MRI, NeuroImage Clin., 22 (2019), 101748. https://doi.org/10.1016/j.nicl.2019.101748 doi: 10.1016/j.nicl.2019.101748
    [31] S. L. Oh, Y. Hagiwara, U. Raghavendra, R. Yuvaraj, N. Arunkumar, M. Murugappan, et al., A deep learning approach for Parkinson's disease diagnosis from EEG signals, Neural Comput. Appl., 32 (2020), 10927–10933. https://doi.org/10.1007/s00521-018-3689-5 doi: 10.1007/s00521-018-3689-5
    [32] J. C. Vasquez-Correa, T. Arias-Vergara, J. R. Orozco-Arroyave, B. M. Eskofier, J. Klucken, E. Noth, Multimodal assessment of Parkinson's disease: a deep learning approach, IEEE J. Biomed. Health Inform., 23 (2019), 1618–1630. https://doi.org/10.1109/JBHI.2018.2866873 doi: 10.1109/JBHI.2018.2866873
    [33] A. Talitckii, E. Kovalenko, A. Anikina, O. Zimniakova, M. Semenov, E. Bril, et al., Avoiding misdiagnosis of Parkinson's disease with the use of wearable sensors and artificial intelligence, IEEE Sensors J., 21 (2021), 3738–3747. https://doi.org/10.1109/JSEN.2020.3027564 doi: 10.1109/JSEN.2020.3027564
    [34] C. R. Pereira, D. R. Pereira, F. A. da Silva, C. Hook, S. A. Weber, L. A. M. Pereira, et al., A step towards the automated diagnosis of Parkinson's disease: analyzing handwriting movements, 2015 IEEE 28th International Symposium on Computer-Based Medical Systems, 2015,171–176. https://doi.org/10.1109/CBMS.2015.34 doi: 10.1109/CBMS.2015.34
    [35] C. R. Pereira, D. R. Pereira, J. P. Papa, G. H. Rosa, X. S. Yang, Convolutional neural networks applied for Parkinson's disease identification, In: A. Holzinger, Machine learning for health informatics. lecture notes in computer science, Cham: Springer, 9605 (2016), 377–390. https://doi.org/10.1007/978-3-319-50478-0_19
    [36] C. R. Pereira, D. R. Pereira, G. H. Rosa, V. H. Albuquerque, S. A. Weber, C. Hook, et al., Handwritten dynamics assessment through convolutional neural networks: an application to Parkinson's disease identification, Artif. Intell. Med., 87 (2018), 67–77. https://doi.org/10.1016/j.artmed.2018.04.001 doi: 10.1016/j.artmed.2018.04.001
    [37] M. Shaban, Deep convolutional neural network for Parkinson's disease based handwriting screening, 2020 IEEE 17th International Symposium on Biomedical Imaging Workshops (ISBI Workshops), 2020, 1–4. https://doi.org/10.1109/ISBIWorkshops50223.2020.9153407 doi: 10.1109/ISBIWorkshops50223.2020.9153407
    [38] T. J. Wroge, Y. Ozkanca, C. Demiroglu, D. Si, D. C. Atkins, R. H. Ghomi, Parkinson's disease diagnosis using machine learning and voice, 2018 IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2018, 1–7. https://doi.org/10.1109/SPMB.2018.8615607 doi: 10.1109/SPMB.2018.8615607
    [39] Y. Dai, Z. Tang, Y. Wang, Z. Xu, Data driven intelligent diagnostics for Parkinson's disease, IEEE Access, 7 (2019), 106941–106950. https://doi.org/10.1109/ACCESS.2019.2931744 doi: 10.1109/ACCESS.2019.2931744
    [40] J. Rusz, M. Novotny, J. Hlavnicka, T. Tykalova, E. Ruzicka, High-accuracy voice-based classification between patients with Parkinson's disease and other neurological diseases may be an easy task with inappropriate experimental design, IEEE Trans. Neural Syst. Rehabil. Eng., 25 (2017), 1319–1321. https://doi.org/10.1109/TNSRE.2016.2621885 doi: 10.1109/TNSRE.2016.2621885
    [41] A. U. Haq, J. P. Li, M. H. Memon, J. Khan, A. Malik, T. Ahmad, et al., Feature selection based on L1-norm support vector machine and effective recognition system for Parkinson's disease using voice recordings, IEEE Access, 7 (2019), 37718–37734. https://doi.org/10.1109/ACCESS.2019.2906350 doi: 10.1109/ACCESS.2019.2906350
    [42] J. Prince, F. Andreotti, M. De Vos, Multi-source ensemble learning for the remote prediction of Parkinson's disease in the presence of source-wise missing data, IEEE Trans. Biomed. Eng., 66 (2019), 1402–1411. https://doi.org/10.1109/TBME.2018.2873252 doi: 10.1109/TBME.2018.2873252
    [43] W. Zeng, F. Liu, Q. Wang, Y. Wang, L. Ma, Y. Zhang, Parkinson's disease classification using gait analysis via deterministic learning, Neurosci. Lett., 633 (2016), 268–278. https://doi.org/10.1016/j.neulet.2016.09.043 doi: 10.1016/j.neulet.2016.09.043
    [44] A. Muniz, H. Liu, K. Lyons, R. Pahwa, W. Liu, F. Nobre, et al., Comparison among probabilistic neural network, support vector machine and logistic regression for evaluating the effect of subthalamic stimulation in Parkinson disease on ground reaction force during gait, J. Biomech., 43 (2010), 720–726. https://doi.org/10.1016/j.jbiomech.2009.10.018 doi: 10.1016/j.jbiomech.2009.10.018
    [45] F. M. J. Pfister, T. T. Um, D. C. Pichler, J. Goschenhofer, K. Abedinpour, M. Lang, et al., High-resolution motor state detection in Parkinson's disease using convolutional neural networks, Sci. Rep., 100 (2020), 5860. https://doi.org/10.1038/s41598-020-61789-3 doi: 10.1038/s41598-020-61789-3
    [46] P. Drotár, J. Mekyska, I. Rektorová, L. Masarová, Z. Smékal, M. Faundez-Zanuy, Analysis of in-air movement in handwriting: a novel marker for Parkinson's disease, Comput. Methods Programs Biomed., 117 (2014), 405–411. https://doi.org/10.1016/j.cmpb.2014.08.007 doi: 10.1016/j.cmpb.2014.08.007
    [47] M. I. Vanegas, M. F. Ghilardi, S. P. Kelly, A. Blangero, Machine learning for EEG-based biomarkers in Parkinson's disease, 2018 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), 2018, 2661–2665. https://doi.org/10.1109/BIBM.2018.8621498 doi: 10.1109/BIBM.2018.8621498
    [48] S. L. Oh, Y. Hagiwara, U. Raghavendra, R. Yuvaraj, N. Arunkumar, M. Murugappan, et al., A deep learning approach for Parkinson's disease diagnosis from EEG signals, Neural Comput. Appl., 32 (2018), 10927–10933. https://doi.org/10.1007/s00521-018-3689-5 doi: 10.1007/s00521-018-3689-5
    [49] N. Wagh, Y. Varatharajah, EEG-GCNN: augmenting electroencephalogram-based neurological disease diagnosis using a domain-guided graph convolutional neural network, Proc. Mach. Learn. Res., 136 (2020), 367–378.
    [50] X. Shi, T. Wang, L. Wang, H. Liu, N. Yan, Hybrid convolutional recurrent neural networks outperform CNN and RNN in task-state EEG detection for Parkinson's disease, 2019 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC), 2019, 18–21. https://doi.org/10.1109/APSIPAASC47483.2019.9023190 doi: 10.1109/APSIPAASC47483.2019.9023190
    [51] X. Zhang, Y. Yang, H. Wang, S. Ning, H. Wang, Deep neural networks with broad views for Parkinson's disease screening, 2019 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), 2019, 1018–1022. https://doi.org/10.1109/BIBM47256.2019.8983000 doi: 10.1109/BIBM47256.2019.8983000
    [52] V. M. Ramirez, V. Kmetzsch, F. Forbes, M. Dojat, Deep learning models to study the early stages of Parkinson's disease, 2020 IEEE 17th International Symposium on Biomedical Imaging (ISBI), 2020, 1534–1537. https://doi.org/10.1109/ISBI45749.2020.9098529 doi: 10.1109/ISBI45749.2020.9098529
    [53] J. Prasuhn, M. Heldmann, T. F. Münte, N. Brüggemann, A machine learning-based classification approach on Parkinson's disease diffusion tensor imaging datasets, Neurol. Res. Pract., 2 (2020), 46. https://doi.org/10.1186/s42466-020-00092-y doi: 10.1186/s42466-020-00092-y
    [54] J. Rasheed, A. A. Hameed, N. Ajlouni, A. Jamil, A. Ozyavas, Z. Orman, Application of adaptive back-propagation neural networks for Parkinson's disease prediction, 2020 International Conference on Data Analytics for Business and Industry: Way Towards a Sustainable Economy (ICDABI), 2020, 1–5. https://doi.org/10.1109/ICDABI51230.2020.9325709 doi: 10.1109/ICDABI51230.2020.9325709
    [55] H. Gunduz, Deep learning-based Parkinson's disease classification using vocal feature sets, IEEE Access, 7 (2019), 115540–115551. https://doi.org/10.1109/ACCESS.2019.2936564 doi: 10.1109/ACCESS.2019.2936564
    [56] S. Moon, H. J. Song, V. D. Sharma, K. E. Lyons, R. Pahwa, A. E. Akinwuntan, et al., Classification of Parkinson's disease and essential tremor based on balance and gait characteristics from wearable motion sensors via machine learning techniques: a data-driven approach, J. NeuroEng. Rehabil., 17 (2020), 125. https://doi.org/10.1186/s12984-020-00756-5 doi: 10.1186/s12984-020-00756-5
    [57] W. Zeng, F. Liu, Q. Wang, Y. Wang, L. Ma, Y. Zhang, Parkinson's disease classification using gait analysis via deterministic learning, Neurosci. Lett., 633 (2016), 268–278. https://doi.org/10.1016/j.neulet.2016.09.043 doi: 10.1016/j.neulet.2016.09.043
    [58] F. M. J. Pfister, T. T. Um, D. C. Pichler, J. Goschenhofer, K. Abedinpour, M. Lang, et al., High-resolution motor state detection in Parkinson's disease using convolutional neural networks, Sci. Rep., 10 (2020), 5860. https://doi.org/10.1038/s41598-020-61789-3 doi: 10.1038/s41598-020-61789-3
    [59] R. T. White, Classifying Parkinson's disease through image analysis: Part 2, Towards Data Science. Available form: https://towardsdatascience.com/classifying-parkinsons-disease-through-image-analysis-part-2-ddbbf05aac21.
    [60] Parkinsons drawing Pytorch lightning CNN, Kaggle. Available form: https://www.kaggle.com/code/stpeteishii/parkinsons-drawing-pytorch-lightning-cnn.
    [61] M. Shaban, Deep convolutional neural network for Parkinson's disease based handwriting screening, 2020 IEEE 17th International Symposium on Biomedical Imaging Workshops (ISBI Workshops), 2020, 1–4. https://doi.org/10.1109/ISBIWorkshops50223.2020.9153407 doi: 10.1109/ISBIWorkshops50223.2020.9153407
    [62] Detecting Parkinson's disease with OpenCV, computer vision, and the spiral/wave test, Adrian Rosebrock, April 29, 2019. Available form: https://pyimagesearch.com/2019/04/29/detecting-parkinsons-disease-with-opencv-computer-vision-and-the-spiral-wave-test/.
    [63] P. Zham, D. K. Kumar, P. Dabnichki, S. Poosapadi Arjunan, S. Raghav, Distinguishing different stages of Parkinson's disease using composite index of speed and pen-pressure of sketching a spiral, Front Neurol., 8 (2017), 435. https://doi.org/10.3389/fneur.2017.00435 doi: 10.3389/fneur.2017.00435
    [64] J. B. Liu, N. Salamat, M. Kamran, S. Ashraf, R. H. Khan, Single-valued neutrosophic set with quaternion information: a promising approach to assess image quality, Fractals, 31 (2023), 1–10. https://doi.org/10.1142/S0218348X23400741 doi: 10.1142/S0218348X23400741
    [65] J. B. Liu, X. B. Peng, J. Zhao, Analyzing the spatial association of household consumption carbon emission structure based on social network, J. Comb. Optim., 45 (2023), 79. https://doi.org/10.1007/s10878-023-01004-x doi: 10.1007/s10878-023-01004-x
    [66] J. B. Liu, Y. Bao, W. T. Zheng, S. Hayat, Network coherence analysis on a family of nested weighted n-polygon networks, Fractals, 29 (2021), 2150260. https://doi.org/10.1142/S0218348X21502601 doi: 10.1142/S0218348X21502601
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