should be possibly replaced by a sum over paths.

Higher order susceptibility coefficients are built as well in terms of elementary building blocks given by the pairwise susceptibility coefficients $\chi_{ij}$. The notations generalize into the following straightforward manner: $$\begin{aligned}
m_{ijk} &{\stackrel{\mbox{\upshape\tiny def}}{=}}{\mathbb E}(s_is_js_k) {\stackrel{\mbox{\upshape\tiny def}}{=}}m_im_jm_k + m_i\chi_{jk}+m_j\chi_{ik}+m_k\chi_{ij}+\chi_{ijk} \\[0.2cm]
m_{ijkl} &{\stackrel{\mbox{\upshape\tiny def}}{=}}{\mathbb E}(s_is_js_ks_l) {\stackrel{\mbox{\upshape\tiny def}}{=}}m_im_jm_km_l \\[0.2cm] 
&+m_im_j\chi_{kl}+m_im_k\chi_{jl}+m_im_l\chi_{jk}+m_jm_k\chi_{il}+m_jm_l\chi_{ik}+m_km_l\chi_{ij}\\[0.2cm]
&+m_i\chi_{jkl}+m_j\chi_{ikl}+m_k\chi_{ijl}+m_l\chi_{ijk}+\chi_{ijkl}, \end{aligned}$$ where $\chi_{ijk}$ and $\chi_{ijkl}$ are respectively three and four point susceptibilities. The quantities being related to the corresponding marginals similarly to (\[eq:pi\],\[eq:pij\]): $$\begin{aligned}
p(s_i,s_j,s_k) &= \frac{1}{8}\bigl(1+m_is_i+m_js_j+m_ks_k\\[0.2cm]
&+m_{ij}s_is_j+m_{ik}s_is_k+m_{jk}s_js_k+m_{ijk}s_is_js_k\bigr)\\[0.2cm]
p(s_i,s_j,s_k,s_l) &= \frac{1}{16}\bigl(1+m_is_i+m_js_j+m_ks_k+m_ls_l\\[0.2cm]
&+m_{ij}s_is_j+m_{ik}s_is_k+m_{il}s_is_l+m_{jk}s_js_k+m_{jl}s_js_l+m_{kl}s_ks_l\\[0.2cm]
&+m_{ijk}s_is_js_k+m_{ijl}s_is_js_l+m_{ikl}s_is_ks_l+m_{jkl}s_js_ks_l+m_{ijkl}s_is_js_ks_l\bigr)\end{aligned}$$ Using the basic fact that, on the tree $$p(s_i,s_j,s_k) = \frac{p(s_i,s_j)p(s_j,s_k)}{p(s_i)}$$ when $j$ is on the path ${\wideparen{ik}}$ given by ${\mathcal{T}}$, and $$p(s_i,s_j,s_k) = \sum_{s_l}\frac{p(s_i,s_l)p(s_j,s_l)p(s_k,s_l)}{p(s_l)^2}$$ when path $\wideparen{ij}$, $\wideparen{ik}$ and ${\wideparen{jk}}$ along ${\mathcal{T}}$ intersect on vertex $l$, we obtain $$\begin{aligned}
\chi_{ijk} = \begin{cases}
\DD -2\frac{m_l}{(1-m_l^2)^2}\chi_{il}\chi_{jl}\chi_{kl} & \text{with}\ \{l\} =
(i,j)\cap (i,k)\cap(j,k)
\ \text{along}\ {\mathcal{T}},\\[0.2cm]
\DD -2m_j\chi_{ik}& \text{if}\ j\in(i,k)\ \text{along}\ {\mathcal{T}}.
\end{cases}\end{aligned}$$ For the fourth order, more cases have to be distinguished. When $i$, $j$, $k$ and $l$ are aligned as on Figure \[fig:vertex\].c, in this order on the path ${\wideparen{il}}$ along ${\mathcal{T}}$ we have $$p(s_i,s_j,s_k,s_l) = \frac{p(s_i,s_j)p(s_j,s_k,s_l)}{p(s_j)^2}$$ which leads to $$\chi_{ijkl} = 4m_km_j\chi_{il} - \chi_{ik}\chi_{jl}-\chi_{il}\chi_{jk}.$$ When path ${\wideparen{ij}}$, ${\wideparen{ik}}$ and 