Fig. 4.

(A) Portion of a BP factor graph for a replicated variable $W_j$ with $y=3$ replicas and a reference configuration $W_j^{\star }$. The dashed lines represent edges with the rest of the factor graph. The squares represent the interactions $\gamma W_j^{\star }{W}_j^a$. All BP messages (arrows) are assumed to be the same in corresponding edges. (B) Transformed graph that represents the same graph as in A but exploits the symmetry to reduce the number of nodes, keeping only one representative per replica. The hexagon represents a pseudoself-interaction, that is, it expresses the fact that $m_{\star \to j}$ depends on $m_{j\to \star }$ and is parametrized by $\gamma$ and $y$.