Network inference using asynchronously updated kinetic Ising model

Network structures are reconstructed from dynamical data by respectively naive mean field (nMF) and Thouless-Anderson-Palmer (TAP) approximations. TAP approximation adds simple corrections to the nMF approximation, taking into account the effect of the focused spin on itself via its influence on other neighboring spins. For TAP approximation, we use two methods to reconstruct the network: (a) iterative method; (b) casting the inference formula to a set of cubic equations and solving it directly. We investigate inference of the asymmetric Sherrington-Kirkpatrick (aS-K) model using asynchronous update. The solutions of the set of cubic equations depend on temperature $T$ in the aS-K model, and a critical temperature $T_c\approx 2.1$ is found. The two methods for TAP approximation produce the same results when the iterative method is convergent. Compared to nMF, TAP is somewhat better at low temperatures, but approaches the same performance as temperature increases. Both nMF and TAP approximation reconstruct better for longer data length $L$, but for the degree of improvement, TAP performs better than nMF.

Figure 1

The fraction of three real roots for the cubic equation set Eq. (19). A transition seems to occur around $T_c=2.1$. Here we find larger $N$, the transition curve sharper. The parameter values: $\theta =0.5$, $k=1$, $L=20\times {10}^6$.

Figure 2

The reconstruction error $\Delta$ with temperature $T$ for both nMF and TAP approximation. The other parameter values: $N=20$, data length $L=20\times {10}^{10}$, external field $\theta =0$, asymmetric degree $k=1$. Black squares, nMF; red circles, cubic equation method for TAP; blue triangles, iterative method for TAP. Each data point is averaged on 10 realizations.

Figure 3

The scatter plot for the reconstructed couplings versus the true ones. The parameter values: $N=20$, temperature $T=3.7$, external field $\theta =0$, asymmetric degree $k=1$. Black squares, inferred couplings using nMF versus $J_{\mathit{ij}}^{\mathrm{true}}$; blue circles, iterative equation method of TAP versus $J_{\mathit{ij}}^{\mathrm{true}}$; red triangles, cubic method of TAP versus $J_{\mathit{ij}}^{\mathrm{true}}$.

Figure 4

Reconstruction error $\Delta$ versus the data length $L$ for nMF and TAP approximation. The other parameter values: $N=20$, temperature $T=8$, external field $\theta =0$, asymmetric degree $k=1$. Black squares, nMF; blue circles, iterative equation method of TAP; red triangles, cubic method of TAP.