# Free Energy Approximations for CSMA networks

## TL;DR

This paper introduces a novel free energy approximation method for estimating back-off rates in CSMA networks, providing explicit formulas and recursive algorithms, with proven accuracy on certain graph classes and validated through numerical experiments.

## Contribution

It presents a new class of region-based free energy approximations with explicit formulas and recursive methods for CSMA networks, including a special clique approximation and proofs of exactness on chordal graphs.

## Key findings

- The size k_max clique approximation matches Kikuchi free energy.
- The approximation is exact on chordal conflict graphs when k_max = n.
- Numerical experiments show improved accuracy over existing methods.

## Abstract

In this paper we study how to estimate the back-off rates in an idealized CSMA network consisting of $n$ links to achieve a given throughput vector using free energy approximations. More specifically, we introduce the class of region-based free energy approximations with clique belief and present a closed form expression for the back-off rates based on the zero gradient points of the free energy approximation (in terms of the conflict graph, target throughput vector and counting numbers). Next we introduce the size $k_{max}$ clique free energy approximation as a special case and derive an explicit expression for the counting numbers, as well as a recursion to compute the back-off rates. We subsequently show that the size $k_{max}$ clique approximation coincides with a Kikuchi free energy approximation and prove that it is exact on chordal conflict graphs when $k_{max} = n$. As a by-product these results provide us with an explicit expression of a fixed point of the inverse generalized belief propagation algorithm for CSMA networks. Using numerical experiments we compare the accuracy of the novel approximation method with existing methods.