Global hierarchy vs. local structure:
Spurious self-feedback in Barabási-Albert Ising networks
Claudia Merger, Timo Reinartz, Stefan Wessel, Andreas Schuppert, Carsten Honerkamp, Moritz Helias
Barabási-Albert networks
+
Ising model
Barabási-Albert networks
have hubs!
Complicated connectivity
Monte-Carlo
Effective action
Methods
High temperature expansion
"Ground truth"
Monte-Carlo
Methods
"Ground truth"
Problem of "freezing hubs"
Timo Reinartz, Stefan Wessel
Monte-Carlo
"Ground truth"
"conventional" Metropolis Monte Carlo: Hubs freeze out
Parallel Tempering
Swedensen and Wang,1986
\( T \)
\(\beta_i \)
\(\beta_j \)
\( p_{ij} = \min \left( 1, e^{(E_i-E_j)(\beta_i -\beta_j)}\right) \)
Timo Reinartz, Stefan Wessel
Effective action
Methods
High temperature expansion
Mean-field self-consistency
Find transition temperature!
Local structure
\( \rightarrow \) solve for largest eigenvalue of \( A \)
Goh, Kahng, Kim, 2001
"onion structure of eigenvector"
\( N = 10^4 \)
Local structure
\( \rightarrow \) solve for largest eigenvalue of \( A \)
Bianconi, 2002
Goh, Kahng, Kim, 2001
"onion structure of eigenvector"
\( N = 10^4 \)
degreewise parametrization
\( A_{ij} \leftrightarrow p_c(k_i,k_j)=\frac{k_i k_j}{2 m_0 N} \)
Local structure
\( \rightarrow \) solve for largest eigenvalue of \( A \)
degreewise parametrization
\( A_{ij} \leftrightarrow p_c(k_i,k_j)=\frac{k_i k_j}{2 m_0 N} \)
Bianconi, 2002
Goh, Kahng, Kim, 2001
"onion structure of eigenvector"
Monte-Carlo:
Transition Temperature?
Local structure
\( \rightarrow \) solve for largest eigenvalue of \( A \)
degreewise parametrization
"onion structure of eigenvector"
Monte-Carlo:
Onion structure?
\( m_i \sim k_i \)
TAP self-consistency
TAP
mean-field
\( \rightarrow \) solve for special \(A\)
Find transition temperature!
Full TAP solution
\( \rightarrow \) Good general agreement
between Monte-Carlo and TAP
(better than mean-field)
Why is TAP better than local meanfield?
TAP
mean-field
TAP
mean-field
expand \( m_j \) around \( m_i =0 \)
\( m_{i}=\beta K_{0}\sum_{j}\left[A_{ij}\left(\quad\Big|_{m_{i}=0}+m_{i}A_{ji}\beta K_{\text{0}}\right)-\beta K_{0}\delta_{ij}k_{i}m_{i}\right]+ \mathcal{O} (\beta^3 K_0^3).\)
\(m_j \)
\( i\)
\( j\)
TAP
mean-field
expand \( m_j \) around \( m_i =0 \)
\( m_{i}=\beta K_{0}\sum_{j}\left[A_{ij}\left(\quad\Big|_{m_{i}=0}+m_{i}A_{ji}\beta K_{\text{0}}\right)-\beta K_{0}\delta_{ij}k_{i}m_{i}\right]+ \mathcal{O} (\beta^3 K_0^3).\)
\(m_j \)
\( i\)
\( j\)
\( \leftrightarrow \) Mezard, Parisi
and Virasoro, 1978
TAP
mean-field
Local structure
\( \rightarrow \) solve : \( T_T = K_0 \lambda_{B, max} (T_T) \)
degreewise parametrization
insert \( A_{ij} \leftrightarrow p_c(k_i,k_j)=\frac{k_i k_j}{2 m_0 N} \) into
\( m_{i}=\beta K_{0}\underbrace{\sum_{j}A_{ij}m_{j}\Big|_{m_{i}=0}}_{\text{field in the absence of }i\, \approx S} \sim k_i S \)
\( \rightarrow\) Same as meanfield
Global effective field \( S \)
\( S(\beta K_{0})= \frac{1}{\langle k \rangle}\langle k\,m \left(k \right)\rangle_{p(k)} \)
\( m(k)= \tanh\left(\beta K_{0}\,k\,S(\beta K_{0})\right) \)
\( M (S) = \langle \tanh\left(\beta K_{0}\,k\,S(\beta K_{0})\right)\rangle_{p(k)} \)
recover local structure:
\( m_{ NN, h} (k)= \tanh( \beta K_{0} \,(k-1) \, S( \beta K_{0} )+\beta K_{0} m(k_{h}) ) \)
Summary and outlook
- Self-consistent solution without self-feedback
- local "onion" structure plays minor role
Hierarchical nature of connectivity dominates the behaviour
Summary and outlook
- Self-consistent solution without self-feedback
- local "onion" structure plays minor role
Hierarchical nature of connectivity dominates the behaviour
personal takeaway:
- Know what confuses you
- Never give up hope for a simple explanation
Averaging over all configurations
Cumulant generating function
Effective action
deck
By merger
deck
- 104