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||Institute for Theoretical Physics at ETH Zurich
Dynamical Systems in Complex Networks
Lucas BöttcherETH ZürichInstitute for Theoretical PhysicsHIT K 11.3Wolfgang-Pauli-Strasse 278093 Zürich
11.11.2019Lucas Böttcher 1
851-0101-86 SHS 2019
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 2
Dynamical Processes on Networks
The “harmonic oscillator(s)” of network dynamics:
SIS model SIR model
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 3
Dynamical Processes on Networks
Recap SIR model:
SIR model
● Infection dynamics with recovery
● “Standard model” in network dynamics
● Relation between SIR model and percolation
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 4
Order parameter (Site) Percolation
p=pc=0.597...
P(p) |p-p∝c|β
β=5/36 in 2D
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 5
SIR Model
More details on the relation between SIR dynamics and percolation:
Tomé, T., & Ziff, R. M. (2010). Critical behavior of the susceptible-infected-recovered model on a square lattice. Physical Review E, 82(5), 051921.
Newman, M. E. (2002). Spread of epidemic disease on networks. Physical review E, 66(1), 016128.
λ= λc=4.666...
P(λ) |λ-λ∝c|β
β=5/36 in 2D
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 6
Percolation vs. SIR Clusters
Tomé, T., & Ziff, R. M. (2010). Critical behavior of the susceptible-infected-recovered model on a square lattice. Physical Review E, 82(5), 051921.
SIR cluster at λc=4.666...(Bond) percolation cluster at p
c=0.5
origin
Tânia Tomé and Robert M. Ziff
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 7
SIS Model
SIS model
● Infection dynamics without recovery
● “Standard model” in network dynamics
● Relation between SIS model and (directed) percolation
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 8
Directed Percolation
Henkel, M., Hinrichsen, H., Lübeck, S., & Pleimling, M. (2008). Non-equilibrium phase transitions (Vol. 1). Dordrecht: Springer.
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 9
Directed Percolation
pc=0.6447...
P(p) |p-p∝c|β
β=0.580... in 2D
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 10
Contact Process
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 11
Kinetic Monte Carlo
Daniel T. Gillespie (1938-2017)
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 12
Kinetic Monte Carlo
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 13
Kinetic Monte Carlo
λc=1.6488...
P(λ) |λ∝ -λc|β
β=0.580... in 2DP(λ)
λ
Böttcher, L., Woolley-Meza, O., Araújo, N. A., Herrmann, H. J., & Helbing, D. (2015). Disease-induced resource constraints can trigger explosive epidemics. Scientific reports, 5, 16571.
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 14
Kinetic Monte Carlo
SIS dynamics on a square lattice with 642 sites and λ=2
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 15
SIS Model
SIS model
● Infection dynamics without recovery
● “Standard model” in network dynamics
● Relation between SIS model and (directed) percolation
![Page 16: Dynamical Systems in Complex Networks · 2019-11-11 · Dynamical Systems in Complex Networks Lucas Böttcher ETH Zürich Institute for Theoretical Physics HIT K 11.3 Wolfgang-Pauli-Strasse](https://reader033.vdocument.in/reader033/viewer/2022042312/5edaf98909ac2c67fa689bc8/html5/thumbnails/16.jpg)
||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 16
Kinetic Monte Carlo
SIS dynamics on a square lattice with 162 sites and λ=2
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 17
Kinetic Monte Carlo
SIS dynamics on a balanced tree with λ=2
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 18
Kinetic Monte Carlo
SIS dynamics on a Watts-Strogatz graph with λ=2
Stevan Strogatz and Duncan J. Watts
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||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 19
Kinetic Monte Carlo (Non-Poissonian)
Naoki Masuda and Luis E. C. Rocha
Marian Boguñá