

![0_image_0.png](0_image_0.png) 

Figure 8: Two examples of planar graphs used for comparison between methods. We fix the number of concentric polygons to 9 and change the degree d **of the central**
node within the range [3, ..., 25]. (left) Graph for d = 3. (right) **Graph for** d **= 25. Here nodes represent variables and edges pairwise interactions. We also** add external fields which depend on the state of each nodes (not drawn).

increases with the tree-width of the graphs. For large enough N**, exact solution via the**
junction tree method is no longer feasible because of its memory requirements. In contrast, for all approximate inference methods, memory demands do not represent a limitation.

## 3.3 Radial Grid Graphs

In the previous subsection we analyzed the quality of the Z∅ **correction for graphs with a**
regular grid structure. Here, we carry over the analysis of the Z∅ **correction using planar**
graphs which consist of concentric polygons with a variable **number of sides. Figure 8** illustrates these spider-web like graphs. We generate them **as factor graphs with pairwise** interactions which we subsequently convert to an equivalent Forney graph. (See Appendix A for details). Again, local field potentials are parametrized using Θ = 0.**01 and interactions** using β = 1. We analyze the error in Z as a function of the degree d **of the central node.**
Figure 9a shows the median of errors in Z **of 50 random instances. First, we see that**
the variability of all methods, in particular anyTLSBP, is larger than in the regular grid scenario. Also, the improvement of CVM-Loop4 over BP is slightly less significant, possibly caused by the existence of the central node with a large degree. CVM-Loop6 does not converge for instances with d > 4 before 104**seconds and is not included in the analysis.**
We can say that all approaches present results independent of the degree d.

The Z∅ **approximation is the best method compared to the other tested approaches.**
The improvements of Z∅ **on CVM-Loop4 (the second best method) can be of more than**
two orders of magnitude and more than three orders of magnitude compared to BP.