||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

||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

||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

||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

||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

||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

||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

||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.

||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

||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 10

Contact Process

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Kinetic Monte Carlo

Daniel T. Gillespie (1938-2017)

||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 12

Kinetic Monte Carlo

||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.

||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

||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

||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

||Institute for Theoretical Physics at ETH Zurich 11.11.2019Lucas Böttcher 17

Kinetic Monte Carlo

SIS dynamics on a balanced tree with λ=2

||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

||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