Table 1 Graph-based distributed parameter coordination algorithm

Step 0

For all communication nodes, initialize all messages to zero (or close to zero at random), in particular, those from variable nodes to factor

nodes to zero, i.e.,

${\mu }_{p_k\to M_i}^{\left(0\right)}\left(p_k\right)=0$

for all i and k, and set iteration index n=0

Step 1

At each communication node i, receive ${\mu }_{p_k\to M_i}^{\left(n\right)}\left(p_k\right)$ from (the variable node of) each neighbor communication node$k\in {\mathcal{N}}_i$

Step 2

At each communication node i, compute summary messages

${\mu }_{M_i\to p_k}^{(n+1)}\left(p_k\right)=\underset{{\mathbf{p}}_{{\mathcal{A}}_i\setminus \left\{k\right\}}}{min}\left\{M_i\left({\mathbf{p}}_{{\mathcal{A}}_i}\right)+\sum _{j\in {\mathcal{A}}_i\setminus \left\{k\right\}}{\mu }_{p_j\to M_i}^{\left(n\right)}\left(p_j\right)\right\},$

for each neighbor communication node $k\in {\mathcal{A}}_i$, by hypothesizing each possible value of p _k in ${\mathcal{P}}_k$ and finding the best corresponding set of

parameters ${\mathbf{p}}_{{\mathcal{A}}_i\setminus \left\{k\right\}}$ that minimizes the quantity in brackets above

Step 3

At each communication node i, send a table of values representing ${\mu }_{M_i\to p_k}^{(n+1)}\left(p_k\right)$ to (the variable node of) each neighbor communication

node$k\in {\mathcal{N}}_i$

Step 4

At each communication node i, receive a table of values representing ${\mu }_{M_k\to p_i}\left(p_i\right)$ from (the factor node of) each neighbor communication

node$k\in {\mathcal{N}}_i$

Step 5

At each communication node i, generate aggregated messages

${\mu }_{p_i\to M_k}^{(n+1)}\left(p_i\right)={\mu }_{M_i\to p_i}^{(n+1)}\left(p_i\right)+\sum _{j\in {\mathcal{N}}_i\setminus \left\{k\right\}}{\mu }_{M_j\to p_i}^{(n+1)}\left(p_i\right)$

for each neighbor communication node $k\in {\mathcal{A}}_i$ by adding up multiple received tables representing${\left\{{\mu }_{M_j\to p_i}^{(n+1)}\left(p_i\right)\right\}}_{j\in {\mathcal{N}}_i}$

Step 6

At each communication node i, send a table of values representing ${\mu }_{p_i\to M_k}^{(n+1)}\left(p_i\right)$ to (the factor node of) each neighbor communication node

$k\in {\mathcal{N}}_i$

Step 7

Increment iteration index n and go back to step 1 unless a certain stopping criterion, such as reaching a maximum number of iteration, is

satisfied

Step 8

At each communication node i, compute the optimal parameter $p_i^{\ast }$ as

$p_i^{\ast }=\underset{p_i}{argmin}\left\{\sum _{j\in {\mathcal{A}}_i}{\mu }_{M_j\to p_i}^{\left(n_f\right)}\left(p_i\right)\right\},$

where n _f denotes the final value of the iteration index