# Table 1 Graph-based distributed parameter coordination algorithm

Steps

Description

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.,

${\mu }_{{p}_{k}\to {M}_{i}}^{\left(0\right)}\phantom{\rule{0.3em}{0ex}}\left({p}_{k}\right)=0$

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

Step 1

At each communication node i, receive ${\mu }_{{p}_{k}\to {M}_{i}}^{\left(n\right)}\phantom{\rule{0.3em}{0ex}}\left({p}_{k}\right)$ from (the variable node of) each neighbor communication node$k\in {\mathcal{N}}_{i}$

Step 2

At each communication node i, compute summary messages

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

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 ${\mu }_{{M}_{i}\to {p}_{k}}^{\left(n+1\right)}\phantom{\rule{-2.84526pt}{0ex}}\left({p}_{k}\right)$ 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 ${\mu }_{{M}_{k}\to {p}_{i}}\phantom{\rule{-2.84526pt}{0ex}}\left({p}_{i}\right)$ from (the factor node of) each neighbor communication

node$k\in {\mathcal{N}}_{i}$

Step 5

At each communication node i, generate aggregated messages

${\mu }_{{p}_{i}\to {M}_{k}}^{\left(n+1\right)}\left({p}_{i}\right)={\mu }_{{M}_{i}\to {p}_{i}}^{\left(n+1\right)}\left({p}_{i}\right)+\sum _{j\in {\mathcal{N}}_{i}\setminus \left\{k\right\}}{\mu }_{{M}_{j}\to {p}_{i}}^{\left(n+1\right)}\phantom{\rule{-2.84526pt}{0ex}}\left({p}_{i}\right)$

for each neighbor communication node $k\in {\mathcal{A}}_{i}$ by adding up multiple received tables representing${\left\{{\mu }_{{M}_{j}\to {p}_{i}}^{\left(n+1\right)}\phantom{\rule{-2.84526pt}{0ex}}\left({p}_{i}\right)\right\}}_{j\in {\mathcal{N}}_{i}}$

Step 6

At each communication node i, send a table of values representing ${\mu }_{{p}_{i}\to {M}_{k}}^{\left(n+1\right)}\phantom{\rule{-2.84526pt}{0ex}}\left({p}_{i}\right)$ 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

${p}_{i}^{\ast }=\underset{{p}_{i}}{argmin}\left\{\sum _{j\in {\mathcal{A}}_{i}}{\mu }_{{M}_{j}\to {p}_{i}}^{\left({n}_{f}\right)}\phantom{\rule{-2.84526pt}{0ex}}\left({p}_{i}\right)\right\},$

where n f denotes the final value of the iteration index 