SLIDE 1 Predicting synchronization regimes with spectral dimension reduction on graphs
- V. Thibeault, G. St-Onge, X. Roy-Pomerleau, J. G. Young and P. Desrosiers
May 31st, 2019 Département de physique, de génie physique, et d’optique Université Laval, Québec, Canada
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1 Complex systems
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1 Complex systems
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1 Complex systems
SLIDE 5 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk)
SLIDE 6 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk) zj can be complex C
SLIDE 7 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk) zj can be complex C N ≫ 1 coupled equations
SLIDE 8 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk) zj can be complex C N ≫ 1 coupled equations F and G are often nonlinear
SLIDE 9 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk) zj can be complex C N ≫ 1 coupled equations F and G are often nonlinear Ajk = constant ∀j, k ∈ {1, ..., N}
SLIDE 10 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk) zj can be complex C N ≫ 1 coupled equations F and G are often nonlinear Ajk = constant ∀j, k ∈ {1, ..., N} These dynamical systems are often : very hard to analyze mathematically
SLIDE 11 2 Goal
We want to analyze this type of dynamics dzj dt = F(zj) +
N
AjkG(zj, zk) zj can be complex C N ≫ 1 coupled equations F and G are often nonlinear Ajk = constant ∀j, k ∈ {1, ..., N} These dynamical systems are often : very hard to analyze mathematically quite long to integrate numerically Possible solution : Reduce the number of dimensions of the dynamical system.
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3
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3
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3
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3
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3
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4
We don’t want to lose the graph properties by doing the dimension reduction.
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We don’t want to lose the graph properties by doing the dimension reduction. Let’s use the spectral graph theory to find M !
See [Laurence et al, Spectral Dimension Reduction of Complex Dynamical Networks, Phys. Rev. X (2019)]
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5 Spectral weight matrix M
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5 Spectral weight matrix M
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5 Spectral weight matrix M
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5 Spectral weight matrix M
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6
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7
Can we predict synchronization regimes with the spectral dimension reduction?
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8 Synchronization predictions
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8 Synchronization predictions
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8 Synchronization predictions
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8 Synchronization predictions
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9 Bonus
Decipher the influence of the SBM on synchronization
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10 Synchronization in the Cowan-Wilson model
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10 Synchronization in the Cowan-Wilson model
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10 Synchronization in the Cowan-Wilson model
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10 Synchronization in the Cowan-Wilson model
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10 Synchronization in the Cowan-Wilson model
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11 Summary
Spectral graph theory allows to reduce successfully multiple synchronization dynamics
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11 Summary
Spectral graph theory allows to reduce successfully multiple synchronization dynamics Detectability of the SBM delimits synchronization regions in the Cowan-Wilson dynamics
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11 Summary
Spectral graph theory allows to reduce successfully multiple synchronization dynamics Detectability of the SBM delimits synchronization regions in the Cowan-Wilson dynamics Typical Netsci message : Structure influences the dynamics!
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12 Acknowledgements
Thank you! Supervisors : Patrick Desrosiers and Louis J. Dubé Colleagues : Guillaume St-Onge, Xavier Roy-Pomerleau, Charles Murphy, Jean-Gabriel Young, Edward Laurence, Antoine Allard Preprint : Coming soon Contact : vincent.thibeault.1@ulaval.ca
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13 Appendix : Advantages of the spectral dimension reduction
Dimension reduction in synchronization Watanabe-Strogatz (1993) Ott-Antonsen (2008) Spectral(2018-2019) ***original approach*** Advantages of the spectral dimension reduction : N < ∞ Systematic reduction of dynamics on graphs Few hypothesis Not restricted to synchronization dynamics
