Acknowledgements
——————————————————————————————
This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550- 30 22-1-0007
——————————————————————————————
Pipeline
Pipeline
Pipeline
Pipeline
Pipeline
Motivation - Ordinal Partition Networks (OPN)
Motivation - OPN Lorenz System
Lorenz System:
dxdt=σ(y−x),dydt=x(ρ−z),dzdt=xy−βz
Timeseries: x(t)
Periodic: σ=10.0, β=83, ρ=100
Motivation - OPN Lorenz System
Lorenz System:
dxdt=σ(y−x),dydt=x(ρ−z),dzdt=xy−βz
Timeseries: x(t)
Periodic: σ=10.0, β=83, ρ=100
- Chaotic: σ=10.0, β=83, ρ=105
Motivation - OPN with Noise
Periodic Rossler System
dxdt=−y−z,dydt=x+ay,dzdt=b+z(x−c)
a=0.1, b=0.2, c=14
Timeseries: z(t)
Motivation - OPN with Noise
Periodic Rossler System
dxdt=−y−z,dydt=x+ay,dzdt=b+z(x−c)
a=0.1, b=0.2, c=14
Timeseries: z(t)
Motivation - OPN with Noise
Periodic Rossler System
dxdt=−y−z,dydt=x+ay,dzdt=b+z(x−c)
a=0.1, b=0.2, c=14
Timeseries: z(t)
Motivation - OPN with Noise
Periodic Rossler System
dxdt=−y−z,dydt=x+ay,dzdt=b+z(x−c)
a=0.1, b=0.2, c=14
Timeseries: z(t)
Motivation - Undesired Transitions
Coarse-grained State Space Networks (CGSSN)
Network Formulation
Network Formulation
Network Formulation
Network Formulation
Network Formulation
Network Formulation
Network Formulation
Network Formulation
CGSSN with Noise
Periodic Rossler System
dxdt=−y−z,dydt=x+ay,dzdt=b+z(x−c)
a=0.1
- b=0.2
c=14
Timeseries: z(t)
CGSSN with Noise
Periodic Rossler System
dxdt=−y−z,dydt=x+ay,dzdt=b+z(x−c)
a=0.1
- b=0.2
c=14
Timeseries: z(t)
Distance Matrix
Graph Dissimilarity Measures - Shortest Path
- Shortest Path: C(P)=\sum_{e\in P}w(e)
D(a,b)=\min_{P}C(P)
- Unweighted: w(e)=1~\forall~e\in P
Graph Dissimilarity Measures - Shortest Path
- Shortest Path: C(P)=\sum_{e\in P}w(e)
D(a,b)=\min_{P}C(P)
- Unweighted: w(e)=1~\forall~e\in P
Graph Dissimilarity Measures - Shortest Path
- Shortest Path: C(P)=\sum_{e\in P}w(e)
D(a,b)=\min_{P}C(P)
- Unweighted: w(e)=1~\forall~e\in P
Graph Dissimilarity Measures - Weighted Shortest Path
- Weighted Shortest Path: C'(P)=\sum_{e\in P}\frac{1}{w(e)}
D(a,b)=\sum_{e\in P}w(e)
- Shortest Weighted Path: D(a,b)=|\min_P C'(P)|
Graph Dissimilarity Measures - Weighted Shortest Path
- Weighted Shortest Path: C'(P)=\sum_{e\in P}\frac{1}{w(e)}
D(a,b)=\sum_{e\in P}w(e)
- Shortest Weighted Path: D(a,b)=|\min_P C'(P)|
Graph Dissimilarity Measures - Weighted Shortest Path
- Weighted Shortest Path: C'(P)=\sum_{e\in P}\frac{1}{w(e)}
D(a,b)=\sum_{e\in P}w(e)
- Shortest Weighted Path: D(a,b)=|\min_P C'(P)|
Graph Dissimilarity Measures - Weighted Shortest Path
- Weighted Shortest Path: C'(P)=\sum_{e\in P}\frac{1}{w(e)}
D(a,b)=\sum_{e\in P}w(e)
- Shortest Weighted Path: D(a,b)=|\min_P C'(P)|
Graph Dissimilarity Measures - Diffusion Distance
Diffusion Distance: P(i,j)=\frac{A(i,j)}{\sum_{k=1}^{|V|}A(i,k)}
- A: Adjacency matrix with no self loops
- P: Probability of transitioning from i to j
t-random walk: P^t(a,b)
Lazy Transition Probability: \widetilde{\mathbf{P}} = \frac{1}{2}\left[\mathbf{P}(a, b) + \mathbf{I}\right]
d_t(a,b) = \sqrt{ \sum_{c \in V} \frac{1}{\mathbf{d}(c)} { \left[ \widetilde{\mathbf{P}}^t (a,c) - \widetilde{\mathbf{P}}^t (b,c) \right] }^2 }
Graph Dissimilarity Measures - Diffusion Distance
Diffusion Distance: P(i,j)=\frac{A(i,j)}{\sum_{k=1}^{|V|}A(i,k)}
- A: Adjacency matrix with no self loops
- P: Probability of transitioning from i to j
t-random walk: P^t(a,b)
Lazy Transition Probability: \widetilde{\mathbf{P}} = \frac{1}{2}\left[\mathbf{P}(a, b) + \mathbf{I}\right]
d_t(a,b) = \sqrt{ \sum_{c \in V} \frac{1}{\mathbf{d}(c)} { \left[ \widetilde{\mathbf{P}}^t (a,c) - \widetilde{\mathbf{P}}^t (b,c) \right] }^2 }
Graph Dissimilarity Measures - Diffusion Distance
Diffusion Distance: P(i,j)=\frac{A(i,j)}{\sum_{k=1}^{|V|}A(i,k)}
- A: Adjacency matrix with no self loops
- P: Probability of transitioning from i to j
t-random walk: P^t(a,b)
Lazy Transition Probability: \widetilde{\mathbf{P}} = \frac{1}{2}\left[\mathbf{P}(a, b) + \mathbf{I}\right]
d_t(a,b) = \sqrt{ \sum_{c \in V} \frac{1}{\mathbf{d}(c)} { \left[ \widetilde{\mathbf{P}}^t (a,c) - \widetilde{\mathbf{P}}^t (b,c) \right] }^2 }
Graph Dissimilarity Measures - Diffusion Distance
Diffusion Distance: P(i,j)=\frac{A(i,j)}{\sum_{k=1}^{|V|}A(i,k)}
- A: Adjacency matrix with no self loops
- P: Probability of transitioning from i to j
t-random walk: P^t(a,b)
Lazy Transition Probability: \widetilde{\mathbf{P}} = \frac{1}{2}\left[\mathbf{P}(a, b) + \mathbf{I}\right]
d_t(a,b) = \sqrt{ \sum_{c \in V} \frac{1}{\mathbf{d}(c)} { \left[ \widetilde{\mathbf{P}}^t (a,c) - \widetilde{\mathbf{P}}^t (b,c) \right] }^2 }
Graph Dissimilarity Measures - Diffusion Distance
Diffusion Distance: P(i,j)=\frac{A(i,j)}{\sum_{k=1}^{|V|}A(i,k)}
- A: Adjacency matrix with no self loops
- P: Probability of transitioning from i to j
t-random walk: P^t(a,b)
Lazy Transition Probability: \widetilde{\mathbf{P}} = \frac{1}{2}\left[\mathbf{P}(a, b) + \mathbf{I}\right]
d_t(a,b) = \sqrt{ \sum_{c \in V} \frac{1}{\mathbf{d}(c)} { \left[ \widetilde{\mathbf{P}}^t (a,c) - \widetilde{\mathbf{P}}^t (b,c) \right] }^2 }
Network Distances Summary
Persistent Homology
Persistence of Networks
Persistence of Networks
Persistence of Networks
Persistence of Networks
Persistence of Networks
Persistence of Networks
Persistence of Networks
Persistence of Networks
Sine Wave Example
State Identification
Empirical Testing of 23 Dynamical Systems
Empirical Testing of 23 Dynamical Systems
Empirical Testing of 23 Dynamical Systems
Empirical Testing of 23 Dynamical Systems
Empirical Testing of 23 Dynamical Systems
Empirical Testing of 23 Dynamical Systems
Dynamic State Identification Results
Experimental Results - Periodic
Experimental Results - Periodic
Experimental Results - Periodic
Myers, Audun, and Firas A. Khasawneh. "Dynamic State Analysis of a Driven Magnetic Pendulum Using Ordinal Partition Networks and Topological Data Analysis."
Experimental Results - Chaotic
Experimental Results - Chaotic
Experimental Results - Chaotic
Myers, Audun, and Firas A. Khasawneh. "Dynamic State Analysis of a Driven Magnetic Pendulum Using Ordinal Partition Networks and Topological Data Analysis."
Thank you!
Any questions?
Myers, A. D., Chumley, M. M., Khasawneh, F. A., & Munch, E. (2023). Persistent homology of coarse-grained state-space networks. Physical Review E, 107(3), 034303.