Detecting phase transitions in collective behavior using manifold's curvature
If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-...
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| Format: | Article |
| Language: | English |
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AIMS Press
2017-03-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2017027 |
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| _version_ | 1832590038748102656 |
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| author | Kelum Gajamannage Erik M. Bollt |
| author_facet | Kelum Gajamannage Erik M. Bollt |
| author_sort | Kelum Gajamannage |
| collection | DOAJ |
| description | If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples. |
| format | Article |
| id | doaj-art-3ca25563ab9741b5baf5f09e770126eb |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2017-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-3ca25563ab9741b5baf5f09e770126eb2025-01-24T02:39:37ZengAIMS PressMathematical Biosciences and Engineering1551-00182017-03-0114243745310.3934/mbe.2017027Detecting phase transitions in collective behavior using manifold's curvatureKelum Gajamannage0Erik M. Bollt1Department of Mathematics, Clarkson University, Potsdam, NY-13699, USADepartment of Mathematics, Clarkson University, Potsdam, NY-13699, USAIf a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.https://www.aimspress.com/article/doi/10.3934/mbe.2017027phase transitionmanifoldcollective behaviordimensionality reductioncurvature |
| spellingShingle | Kelum Gajamannage Erik M. Bollt Detecting phase transitions in collective behavior using manifold's curvature Mathematical Biosciences and Engineering phase transition manifold collective behavior dimensionality reduction curvature |
| title | Detecting phase transitions in collective behavior using manifold's curvature |
| title_full | Detecting phase transitions in collective behavior using manifold's curvature |
| title_fullStr | Detecting phase transitions in collective behavior using manifold's curvature |
| title_full_unstemmed | Detecting phase transitions in collective behavior using manifold's curvature |
| title_short | Detecting phase transitions in collective behavior using manifold's curvature |
| title_sort | detecting phase transitions in collective behavior using manifold s curvature |
| topic | phase transition manifold collective behavior dimensionality reduction curvature |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2017027 |
| work_keys_str_mv | AT kelumgajamannage detectingphasetransitionsincollectivebehaviorusingmanifoldscurvature AT erikmbollt detectingphasetransitionsincollectivebehaviorusingmanifoldscurvature |