# Subadditivity Beyond Trees and the Chi-Squared Mutual Information

## TL;DR

This paper extends the subadditivity of mutual information to series-parallel graphs with asymmetric channels using Chi-squared mutual information, broadening the understanding of information flow in complex networks.

## Contribution

It demonstrates that subadditivity holds for asymmetric channels on series-parallel graphs using Chi-squared mutual information, a novel approach compared to previous symmetric channel results.

## Key findings

- Subadditivity applies to asymmetric channels on series-parallel graphs.
- Chi-squared mutual information is effective for proving subadditivity in this context.
- The new bounds can be tighter than existing SDPI-based bounds.

## Abstract

In 2000, Evans et al. [Eva+00] proved the subadditivity of the mutual information in the broadcasting on tree model with binary vertex labels and symmetric channels. They raised the question of whether such subadditivity extends to loopy graphs in some appropriate way. We recently proposed such an extension that applies to general graphs and binary vertex labels [AB18], using synchronization models and relying on percolation bounds. This extension requires however the edge channels to be symmetric on the product of the adjacent spins. A more general version of such a percolation bound that applies to asymmetric channels is also obtained in [PW18], relying on the SDPI, but the subadditivity property does not follow with such generalizations. In this note, we provide a new result showing that the subadditivity property still holds for arbitrary (asymmetric) channels acting on the product of spins, when the graphs are restricted to be series-parallel. The proof relies on the use of the Chi-squared mutual information rather than the classical mutual information, and various properties of the former are discussed. We also present a generalization of the broadcasting on tree model (the synchronization on tree) where the bound from [PW18] relying on the SPDI can be significantly looser than the bound resulting from the Chi-squared subadditivity property presented here.