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Global stability of a multistrain SIS model with superinfection
Detecting phase transitions in collective behavior using manifold's curvature
Department of Mathematics, Clarkson University, Potsdam, NY-13699, USA |
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.
References:
[1] |
Birds flying away, shutterstock. Available from: https://www.shutterstock.com/video/clip-3003274-stock-footage-birds-flying-away.html?src=search/Yg-XYej1Po2F0VO3yykclw:1:19/gg. |
[2] |
Data set of detection of unusual crowd activity available at robotics and vision laboratory, Department of Computer Science and Engineering, University of Minnesota. Available from: http://mha.cs.umn.edu/proj_events.shtml. |
[3] |
Data set of pet2009 at Computational Vision Group, University of Reading, 2009. Available from: http://ftp.pets.reading.ac.uk/pub/. |
[4] |
N. Abaid, E. Bollt and M. Porfiri,
Topological analysis of complexity in multiagent systems, Physical Review E, 85 (2012), 041907.
doi: 10.1103/PhysRevE.85.041907. |
[5] |
G. Alfred,
Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC press, 1998. |
[6] |
I. R. de Almeida and C. R. Jung, Change detection in human crowds, in Graphics, Patterns
and Images (SIBGRAPI), 2013 26th SIBGRAPI-Conference on, IEEE, (2013), 63-69.
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M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini and M. Viale,
Empirical investigation of starling flocks: A benchmark study in collective animal behavior, Animal Behaviour, 76 (2008), 201-215.
doi: 10.1016/j.anbehav.2008.02.004. |
[9] |
C. Becco, N. Vandewalle, J. Delcourt and P. Poncin,
Experimental evidences of a structural and dynamical transition in fish school, Physica A: Statistical Mechanics and its Applications, 367 (2006), 487-493.
doi: 10.1016/j.physa.2005.11.041. |
[10] |
M. Beekman, D. J. T. Sumpter and F. L. W. Ratnieks,
Phase transition between disordered and ordered foraging in pharaoh's ants, Proceedings of the National Academy of Sciences, 98 (2001), 9703-9706.
doi: 10.1073/pnas.161285298. |
[11] |
A. C. Bovik,
Handbook of Image and Video Processing, Academic press, 2010. |
[12] |
R. Bracewell,
Fourier Analysis and Imaging, Springer Science & Business Media, 2010.
doi: 10.1007/978-1-4419-8963-5. |
[13] |
I. D. Couzin,
Collective cognition in animal groups, Trends in cognitive sciences, 13 (2009), 36-43.
doi: 10.1016/j.tics.2008.10.002. |
[14] |
I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin,
Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
[15] |
A. Deutsch,
Principles of biological pattern formation: Swarming and aggregation viewed as self organization phenomena, Journal of Biosciences, 24 (1999), 115-120.
doi: 10.1007/BF02941115. |
[16] |
J. H. Friedman, J. L. Bentley and R. A. Finkel,
An algorithm for finding best matches in logarithmic expected time, ACM Transactions on Mathematical Software, 3 (1977), 209-226.
doi: 10.1145/355744.355745. |
[17] |
K. Gajamannage, S. Butailb, M. Porfirib and E. M. Bollt,
Model reduction of collective motion by principal manifolds, Physica D: Nonlinear Phenomena, 291 (2015), 62-73.
doi: 10.1016/j.physd.2014.09.009. |
[18] |
K. Gajamannage, S. Butailb, M. Porfirib and E. M. Bollt,
Identifying manifolds underlying group motion in Vicsek agents, The European Physical Journal Special Topics, 224 (2015), 3245-3256.
doi: 10.1140/epjst/e2015-50088-2. |
[19] |
J. J. Gerbrands,
On the relationships between SVD, KLT and PCA, Pattern recognition, 14 (1981), 375-381.
doi: 10.1016/0031-3203(81)90082-0. |
[20] |
R. Gerlai,
High-throughput behavioral screens: The first step towards finding genes involved in vertebrate brain function using zebra fish, Molecules, 15 (2010), 2609-2622.
doi: 10.3390/molecules15042609. |
[21] |
G. H. Golub and C. Reinsch,
Singular value decomposition and least squares solutions, Numerische Mathematik, 14 (1970), 403-420.
doi: 10.1007/BF02163027. |
[22] |
D. Helbing, J. Keltsch and P. Molnar,
Modelling the evolution of human trail systems, Nature, 388 (1997), 47-50.
|
[23] |
J. M. Lee,
Riemannian Manifolds: An Introduction to Curvature, volume 176, Springer, 1997.
doi: 10.1007/b98852. |
[24] |
J. M. Lee,
Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[25] |
R. Mehran, A. Oyama and M. Shah, Abnormal crowd behavior detection using social force
model, in Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference
on, IEEE, (2009), 935-942.
doi: 10.1109/CVPR.2009.5206641. |
[26] |
M. M. Millonas,
Swarms, Phase Transitions, and Collective Intelligence, Technical report, Los Alamos National Lab., New Mexico, USA, 1992. |
[27] |
S. R. Musse and D. Thalmann, A model of human crowd behavior: Group inter-relationship
and collision detection analysis, in Computer Animation and Simulation, Springer, (1997),
39-51.
doi: 10.1007/978-3-7091-6874-5_3. |
[28] |
M. Nagy, Z. Ákos, D. Biro and T. Vicsek,
Hierarchical group dynamics in pigeon flocks, Nature, 464 (2010), 890-893.
doi: 10.1038/nature08891. |
[29] |
B. O'neill, Elementary Differential Geometry, Academic press, New York, 1966.
![]() ![]() |
[30] |
T. Papenbrock and T. H. Seligman, Invariant manifolds and collective motion in many-body systems, AIP Conf. Proc. , 597(2001), p301, arXiv: nlin/0206035.
doi: 10.1063/1.1427476. |
[31] |
B. L. Partridge,
The structure and function of fish schools, Scientific American, 246 (1982), 114-123.
doi: 10.1038/scientificamerican0682-114. |
[32] |
W. Rappel, A. Nicol, A. Sarkissian, H. Levine and W. F. Loomis,
Self-organized vortex state
in two-dimensional dictyostelium dynamics, Physical Review Letters, 83 (1999), p1247.
|
[33] |
E. M. Rauch, M. M. Millonas and D. R. Chialvo,
Pattern formation and functionality in swarm models, Physics Letters A, 207 (1995), 185-193.
doi: 10.1016/0375-9601(95)00624-C. |
[34] |
V. Y. Rovenskii,
Topics in Extrinsic Geometry of Codimension-one Foliations, Springer, 2011.
doi: 10.1007/978-1-4419-9908-5. |
[35] |
S. T. Roweis and L. K. Saul,
Nonlinear dimensionality reduction by locally linear embedding, Science, 290 (2000), 2323-2326.
doi: 10.1126/science.290.5500.2323. |
[36] |
R. V. Solé, S. C. Manrubia, B. Luque, J. Delgado and J. Bascompte,
Phase transitions and complex systems: Simple, nonlinear models capture complex systems at the edge of chaos, Complexity, 1 (1996), 13-26.
|
[37] |
D. Somasundaram,
Differential Geometry: A First Course, Alpha Science Int'l Ltd., 2005. |
[38] |
D. Sumpter, J. Buhl, D. Biro and I. Couzin,
Information transfer in moving animal groups, Theory in Biosciences, 127 (2008), 177-186.
doi: 10.1007/s12064-008-0040-1. |
[39] |
J. B. Tenenbaum, V. De Silva and J. C. Langford,
A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), 2319-2323.
doi: 10.1126/science.290.5500.2319. |
[40] |
E. Toffin, D. D. Paolo, A. Campo, C. Detrain and J. Deneubourg,
Shape transition during nest digging in ants, Proceedings of the National Academy of Sciences, 106 (2009), 18616-18620.
doi: 10.1073/pnas.0902685106. |
[41] |
C. M. Topaz and A. L. Bertozzi,
Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM Journal on Applied Mathematics, 65 (2004), 152-174.
doi: 10.1137/S0036139903437424. |
[42] |
T. Vicsek, A. Cziró, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[43] |
E. Witten,
Phase transitions in m-theory and f-theory, Nuclear Physics B, 471 (1996), 195-216.
doi: 10.1016/0550-3213(96)00212-X. |
[44] |
P. N. Yianilos, Data structures and algorithms for nearest neighbor search in general metric
spaces, in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms,
Society for Industrial and Applied Mathematics, (1993), 311-321. |
[45] |
T. Zhao and R. Nevatia, Tracking multiple humans in crowded environment, in Computer
Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer
Society Conference on, IEEE, (2004), Ⅱ-406.
doi: 10.1109/CVPR.2004.1315192. |
show all references
References:
[1] |
Birds flying away, shutterstock. Available from: https://www.shutterstock.com/video/clip-3003274-stock-footage-birds-flying-away.html?src=search/Yg-XYej1Po2F0VO3yykclw:1:19/gg. |
[2] |
Data set of detection of unusual crowd activity available at robotics and vision laboratory, Department of Computer Science and Engineering, University of Minnesota. Available from: http://mha.cs.umn.edu/proj_events.shtml. |
[3] |
Data set of pet2009 at Computational Vision Group, University of Reading, 2009. Available from: http://ftp.pets.reading.ac.uk/pub/. |
[4] |
N. Abaid, E. Bollt and M. Porfiri,
Topological analysis of complexity in multiagent systems, Physical Review E, 85 (2012), 041907.
doi: 10.1103/PhysRevE.85.041907. |
[5] |
G. Alfred,
Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC press, 1998. |
[6] |
I. R. de Almeida and C. R. Jung, Change detection in human crowds, in Graphics, Patterns
and Images (SIBGRAPI), 2013 26th SIBGRAPI-Conference on, IEEE, (2013), 63-69.
doi: 10.1109/SIBGRAPI.2013.18. |
[8] |
M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini and M. Viale,
Empirical investigation of starling flocks: A benchmark study in collective animal behavior, Animal Behaviour, 76 (2008), 201-215.
doi: 10.1016/j.anbehav.2008.02.004. |
[9] |
C. Becco, N. Vandewalle, J. Delcourt and P. Poncin,
Experimental evidences of a structural and dynamical transition in fish school, Physica A: Statistical Mechanics and its Applications, 367 (2006), 487-493.
doi: 10.1016/j.physa.2005.11.041. |
[10] |
M. Beekman, D. J. T. Sumpter and F. L. W. Ratnieks,
Phase transition between disordered and ordered foraging in pharaoh's ants, Proceedings of the National Academy of Sciences, 98 (2001), 9703-9706.
doi: 10.1073/pnas.161285298. |
[11] |
A. C. Bovik,
Handbook of Image and Video Processing, Academic press, 2010. |
[12] |
R. Bracewell,
Fourier Analysis and Imaging, Springer Science & Business Media, 2010.
doi: 10.1007/978-1-4419-8963-5. |
[13] |
I. D. Couzin,
Collective cognition in animal groups, Trends in cognitive sciences, 13 (2009), 36-43.
doi: 10.1016/j.tics.2008.10.002. |
[14] |
I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin,
Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516.
doi: 10.1038/nature03236. |
[15] |
A. Deutsch,
Principles of biological pattern formation: Swarming and aggregation viewed as self organization phenomena, Journal of Biosciences, 24 (1999), 115-120.
doi: 10.1007/BF02941115. |
[16] |
J. H. Friedman, J. L. Bentley and R. A. Finkel,
An algorithm for finding best matches in logarithmic expected time, ACM Transactions on Mathematical Software, 3 (1977), 209-226.
doi: 10.1145/355744.355745. |
[17] |
K. Gajamannage, S. Butailb, M. Porfirib and E. M. Bollt,
Model reduction of collective motion by principal manifolds, Physica D: Nonlinear Phenomena, 291 (2015), 62-73.
doi: 10.1016/j.physd.2014.09.009. |
[18] |
K. Gajamannage, S. Butailb, M. Porfirib and E. M. Bollt,
Identifying manifolds underlying group motion in Vicsek agents, The European Physical Journal Special Topics, 224 (2015), 3245-3256.
doi: 10.1140/epjst/e2015-50088-2. |
[19] |
J. J. Gerbrands,
On the relationships between SVD, KLT and PCA, Pattern recognition, 14 (1981), 375-381.
doi: 10.1016/0031-3203(81)90082-0. |
[20] |
R. Gerlai,
High-throughput behavioral screens: The first step towards finding genes involved in vertebrate brain function using zebra fish, Molecules, 15 (2010), 2609-2622.
doi: 10.3390/molecules15042609. |
[21] |
G. H. Golub and C. Reinsch,
Singular value decomposition and least squares solutions, Numerische Mathematik, 14 (1970), 403-420.
doi: 10.1007/BF02163027. |
[22] |
D. Helbing, J. Keltsch and P. Molnar,
Modelling the evolution of human trail systems, Nature, 388 (1997), 47-50.
|
[23] |
J. M. Lee,
Riemannian Manifolds: An Introduction to Curvature, volume 176, Springer, 1997.
doi: 10.1007/b98852. |
[24] |
J. M. Lee,
Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218. Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21752-9. |
[25] |
R. Mehran, A. Oyama and M. Shah, Abnormal crowd behavior detection using social force
model, in Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference
on, IEEE, (2009), 935-942.
doi: 10.1109/CVPR.2009.5206641. |
[26] |
M. M. Millonas,
Swarms, Phase Transitions, and Collective Intelligence, Technical report, Los Alamos National Lab., New Mexico, USA, 1992. |
[27] |
S. R. Musse and D. Thalmann, A model of human crowd behavior: Group inter-relationship
and collision detection analysis, in Computer Animation and Simulation, Springer, (1997),
39-51.
doi: 10.1007/978-3-7091-6874-5_3. |
[28] |
M. Nagy, Z. Ákos, D. Biro and T. Vicsek,
Hierarchical group dynamics in pigeon flocks, Nature, 464 (2010), 890-893.
doi: 10.1038/nature08891. |
[29] |
B. O'neill, Elementary Differential Geometry, Academic press, New York, 1966.
![]() ![]() |
[30] |
T. Papenbrock and T. H. Seligman, Invariant manifolds and collective motion in many-body systems, AIP Conf. Proc. , 597(2001), p301, arXiv: nlin/0206035.
doi: 10.1063/1.1427476. |
[31] |
B. L. Partridge,
The structure and function of fish schools, Scientific American, 246 (1982), 114-123.
doi: 10.1038/scientificamerican0682-114. |
[32] |
W. Rappel, A. Nicol, A. Sarkissian, H. Levine and W. F. Loomis,
Self-organized vortex state
in two-dimensional dictyostelium dynamics, Physical Review Letters, 83 (1999), p1247.
|
[33] |
E. M. Rauch, M. M. Millonas and D. R. Chialvo,
Pattern formation and functionality in swarm models, Physics Letters A, 207 (1995), 185-193.
doi: 10.1016/0375-9601(95)00624-C. |
[34] |
V. Y. Rovenskii,
Topics in Extrinsic Geometry of Codimension-one Foliations, Springer, 2011.
doi: 10.1007/978-1-4419-9908-5. |
[35] |
S. T. Roweis and L. K. Saul,
Nonlinear dimensionality reduction by locally linear embedding, Science, 290 (2000), 2323-2326.
doi: 10.1126/science.290.5500.2323. |
[36] |
R. V. Solé, S. C. Manrubia, B. Luque, J. Delgado and J. Bascompte,
Phase transitions and complex systems: Simple, nonlinear models capture complex systems at the edge of chaos, Complexity, 1 (1996), 13-26.
|
[37] |
D. Somasundaram,
Differential Geometry: A First Course, Alpha Science Int'l Ltd., 2005. |
[38] |
D. Sumpter, J. Buhl, D. Biro and I. Couzin,
Information transfer in moving animal groups, Theory in Biosciences, 127 (2008), 177-186.
doi: 10.1007/s12064-008-0040-1. |
[39] |
J. B. Tenenbaum, V. De Silva and J. C. Langford,
A global geometric framework for nonlinear dimensionality reduction, Science, 290 (2000), 2319-2323.
doi: 10.1126/science.290.5500.2319. |
[40] |
E. Toffin, D. D. Paolo, A. Campo, C. Detrain and J. Deneubourg,
Shape transition during nest digging in ants, Proceedings of the National Academy of Sciences, 106 (2009), 18616-18620.
doi: 10.1073/pnas.0902685106. |
[41] |
C. M. Topaz and A. L. Bertozzi,
Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM Journal on Applied Mathematics, 65 (2004), 152-174.
doi: 10.1137/S0036139903437424. |
[42] |
T. Vicsek, A. Cziró, E. Ben-Jacob, I. Cohen and O. Shochet,
Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75 (1995), 1226-1229.
doi: 10.1103/PhysRevLett.75.1226. |
[43] |
E. Witten,
Phase transitions in m-theory and f-theory, Nuclear Physics B, 471 (1996), 195-216.
doi: 10.1016/0550-3213(96)00212-X. |
[44] |
P. N. Yianilos, Data structures and algorithms for nearest neighbor search in general metric
spaces, in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms,
Society for Industrial and Applied Mathematics, (1993), 311-321. |
[45] |
T. Zhao and R. Nevatia, Tracking multiple humans in crowded environment, in Computer
Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer
Society Conference on, IEEE, (2004), Ⅱ-406.
doi: 10.1109/CVPR.2004.1315192. |









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