Claudia Merger, Jasper Albers,
Carsten Honerkamp, Moritz Helias
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
Self-feedback is forbidden!
Self-feedback is forbidden!
Self-feedback is forbidden!
input field \(\theta_i\) for each node \(i\)
expectation values
\(\rho^{S}_i(t)=\langle S_i(t) \rangle \)
\(\rho^{I}_i(t)=\langle I_i(t) \rangle \)
this approximation artificially introduces self-feedback
this approximation artificially introduces self-feedback
We compute second order corrections
Roudi, Y. & Hertz, J. Dynamical TAP equations for non-equilibrium Ising spin glasses. Journal of Statistical Mechanics: Theory and Experiment 2011, P03031 (2011).
Difficulty: Spiking neurons \( \rightarrow \) neurons with 3 discrete states
basis for implementation: binary neurons
idea: communicate changes to \( \theta_i \)
signal change of state via spikes:
Spike with multiplicity 1: \( S \rightarrow I \)
Spike with multiplicity 2: \( I \rightarrow S,R \)
\( \Rightarrow \) Sparsity suppresses activity!
same corrections even though self-feedback is allowed.
only partial cancellation
same corrections even though self-feedback is allowed.
only partial cancellation
State with finite endemic activity exists,
but at which \( \lambda=\frac{\beta}{\mu} \) ?
State with finite endemic activity exists,
but at which \( \lambda=\frac{\beta}{\mu} \) ?
\( \Rightarrow \) Linearize around \( \rho^I =0\) and check when \(\Delta \rho^I >0 \)
State with finite endemic activity exists,
but at which \( \lambda=\frac{\beta}{\mu} \) ?
\( \Rightarrow \) Linearize around \( \rho^I =0\) and check when \(\Delta \rho^I >0 \)
Mean-field: \(\lambda_c^{-1} = \lambda_a^{max} \)
\(a= \) adjacency matr.
State with finite endemic activity exists,
but at which \( \lambda=\frac{\beta}{\mu} \) ?
\( \Rightarrow \) Linearize around \( \rho^I =0\) and check when \(\Delta \rho^I >0 \)
Mean-field: \(\lambda_c^{-1} = \lambda_a^{max} \)
corrected equation:
\( \lambda_c^{-1} = \lambda_D^{max} \)
\(a= \) adjacency matr.
Mean-field: \(\lambda_c^{-1} = \lambda_a^{max} \)
corrected equation:
\( \lambda_c^{-1} = \lambda_D^{max} \)
Mean-field: \(\lambda_c^{-1} = \lambda_a^{max} \)
corrected equation:
\( \lambda_c^{-1} = \lambda_D^{max} \)
corrected equation:
\(\Delta \rho^I = \left( \lambda \lambda_D^{max} -1 \right) \mu \rho^I \)
Imaginary leading eigenvalues \( \lambda_D^{max} \) ?!
Yes, for special network architectures
corrected equation:
\(\Delta \rho^I = \left( \lambda \lambda_D^{max} -1 \right) \mu \rho^I \)
Random Regular Networks with \(\langle k \rangle = 3\)
(all nodes have equal degree, connections random)
corrected equation:
\(\Delta \rho^I = \left( \lambda \lambda_D^{max} -1 \right) \mu \rho^I \)
Random Regular Networks with \(\langle k \rangle = 3\)
(all nodes have equal degree, connections random)
corrected equation:
\(\Delta \rho^I = \left( \lambda \lambda_D^{max} -1 \right) \mu \rho^I \)
time
pop. fraction
Imaginary eigenvalues?!
Yes, for special network architectures
Random Regular Networks
(all nodes have equal degree, connections random)
Open questions and To-Dos
When setting "num_threads" >1:
Change in or degree
Change in inf. dynamics
However: Here, meanfield is done only in population average \( \hat{s}_i \approx \hat{s} \)
Dynamics explained by...
Experiment: Linear Chain, individual-based.
Mean-field overestimates inf. dynamics even if network architecture is accounted for!
More literature on SIR model
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
More information: Pastor-Satorras et. al., A. Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015).
Self-feedback is strongest for hubs!
Castellano C, et. al. Cumulative Merging Percolation and the Epidemic Transition of the Susceptible-Infected-Susceptible Model in Networks. Phys Rev X. 2020 Jan 1;10(1):011070. doi: 10.1103/PhysRevX.10.011070.
mean-field
mean-field
flu. correction / "TAP term"
flu. correction / "TAP term"
mean-field
mean-field
flu. correction / "TAP term"
flu. correction / "TAP term"
Correction cancels self-feedback!
mean-field
mean-field
flu. correction / "TAP term"
flu. correction / "TAP term"
Analogous 2-state problem:
Roudi, Y. & Hertz, J. Dynamical TAP equations for non-equilibrium Ising spin glasses. Journal of Statistical Mechanics: Theory and Experiment 2011, P03031 (2011).
update probability:
Analogous 2-state problem:
Roudi, Y. & Hertz, J. Dynamical TAP equations for non-equilibrium Ising spin glasses. Journal of Statistical Mechanics: Theory and Experiment 2011, P03031 (2011).
update probability:
Cumulant generating functon
Cumulant generating functon
Cumulant generating functon
Cumulant generating functon
Cumulant generating functon
Effective action: Legendre transform
Effective action: Legendre transform
eq. of state
Effective action: Legendre transform
eq. of state
mean-field
flu. correction / "TAP term"
Plefka expansion: Expansion in \( \beta \)