{"title":"Scaling behavior of cross-entropy loss in the identification of percolation phase transitions.","authors":"Huiyao Li, Yu Zhao, Bo Yang","doi":"10.1103/PhysRevE.110.054133","DOIUrl":null,"url":null,"abstract":"<p><p>The cross-entropy loss function is widely used in machine learning to measure the performance of a classification model. Interestingly, our study find that this function has scaling behavior when deep neural networks are used to investigate percolation models. Specifically, we use convolutional neural networks with different pooling methods to study the site percolation on square lattices under two labeling methods (labeling based on spanning cluster and the exact solution of the critical point). Subsequently, graph convolutional neural networks (GCNs) with different pooling methods are utilized to do the same kind of experiment. Finally, the GCN with different pooling methods is used to study the percolation phase transitions on the Erdős-Rényi (ER) networks under labeling based on the critical point. The reliability of the classifiers is detected by the values of the critical point p_{c} and critical exponent ν which are obtained by the scaling behaviors of the percolation probability. The results demonstrate that the scaling exponent of cross-entropy ψ/ν depends on the labeling and pooling methods. Labeling based on critical points, which is equivalent to labeling based on spanning clusters in infinite systems, can be used to investigate the critical behaviors in finite systems. SAGPooling-Mean is an effective pooling method to study the scaling behavior of cross-entropy loss on two-dimensional square lattices and ER networks.</p>","PeriodicalId":20085,"journal":{"name":"Physical review. E","volume":"110 5-1","pages":"054133"},"PeriodicalIF":2.4000,"publicationDate":"2024-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical review. E","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/PhysRevE.110.054133","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Mathematics","Score":null,"Total":0}
Scaling behavior of cross-entropy loss in the identification of percolation phase transitions.
The cross-entropy loss function is widely used in machine learning to measure the performance of a classification model. Interestingly, our study find that this function has scaling behavior when deep neural networks are used to investigate percolation models. Specifically, we use convolutional neural networks with different pooling methods to study the site percolation on square lattices under two labeling methods (labeling based on spanning cluster and the exact solution of the critical point). Subsequently, graph convolutional neural networks (GCNs) with different pooling methods are utilized to do the same kind of experiment. Finally, the GCN with different pooling methods is used to study the percolation phase transitions on the Erdős-Rényi (ER) networks under labeling based on the critical point. The reliability of the classifiers is detected by the values of the critical point p_{c} and critical exponent ν which are obtained by the scaling behaviors of the percolation probability. The results demonstrate that the scaling exponent of cross-entropy ψ/ν depends on the labeling and pooling methods. Labeling based on critical points, which is equivalent to labeling based on spanning clusters in infinite systems, can be used to investigate the critical behaviors in finite systems. SAGPooling-Mean is an effective pooling method to study the scaling behavior of cross-entropy loss on two-dimensional square lattices and ER networks.
期刊介绍:
Physical Review E (PRE), broad and interdisciplinary in scope, focuses on collective phenomena of many-body systems, with statistical physics and nonlinear dynamics as the central themes of the journal. Physical Review E publishes recent developments in biological and soft matter physics including granular materials, colloids, complex fluids, liquid crystals, and polymers. The journal covers fluid dynamics and plasma physics and includes sections on computational and interdisciplinary physics, for example, complex networks.