Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity

TL;DR

This paper analyzes the structure of type graphs, providing sharp bounds on their edge density and biclique regions, and applies these findings to improve bounds in zero-error communication over the binary adder channel.

Contribution

It introduces asymptotically sharp bounds for the maximum edge density and biclique rate region of type graphs, connecting these to noninteractive simulation and hypercontractivity.

Findings

Sharp bounds for maximum edge density of type graphs

Characterization of biclique rate region in type graphs

Improved outer bound for zero-error capacity of binary adder channel

Abstract

In this paper, we study the type graph, namely, a bipartite graph induced by a joint type. We investigate the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale in the length $n$ of the type. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the length of the type goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of $(R_{1},R_{2})$ such that there exists a biclique of the type graph which has respectively $2^{nR_{1}}$ and $2^{nR_{2}}$ vertices on the two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then discuss the connections of these results to noninteractive simulation and hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types and linear algebra.