Influences in Mixing Measures

TL;DR

This paper extends the theory of influences from product measures to more general measures with correlation decay, applying it to voting systems and Markov Random Fields.

Contribution

It develops influence inequalities for measures with correlation decay, generalizing classical results like KKL and Talagrand theorems.

Findings

Analogues of KKL and Talagrand influence theorems for Markov Random Fields

Extension of influence theory to correlated voting scenarios

Insights into functions of spin-systems with correlation decay

Abstract

The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability. This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete groups. Originally, influences of functions were motivated by the study of social choice theory, wherein a Boolean function represents a voting scheme, its inputs represent the votes, and its output represents the outcome of the elections. Thus, product measures represent a scenario in which the votes of the parties are randomly and independently distributed, which is often far from the truth in real-life scenarios. We begin to develop the theory of influences for more general measures under mixing or correlation decay conditions. More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs with correlation decay. We show how some of the original applications of the theory of in terms of voting and coalitions extend to general measures with correlation decay. Our results thus shed light both on voting with correlated voters and on the behavior of general functions of Markov Random Fields (also called ``spin-systems") with correlation decay.