# DeGroot-Friedkin Map in Opinion Dynamics is Mirror Descent

## TL;DR

This paper interprets the DeGroot-Friedkin opinion dynamics map as an entropic mirror descent process, linking social power evolution to information-theoretic optimization on the probability simplex.

## Contribution

It provides a novel variational perspective of the DeGroot-Friedkin map, connecting it to mirror descent and entropy minimization in opinion dynamics.

## Key findings

- DeGroot-Friedkin map is equivalent to an entropic mirror descent.
- Social power aligns with social influence while minimizing extropy.
- Reveals a new optimization-based understanding of opinion formation.

## Abstract

We provide a variational interpretation of the DeGroot-Friedkin map in opinion dynamics. Specifically, we show that the nonlinear dynamics for the DeGroot-Friedkin map can be viewed as mirror descent on the standard simplex with the associated Bregman divergence being equal to the generalized Kullback-Leibler divergence, i.e., an entropic mirror descent. Our results reveal that the DeGroot-Friedkin map elicits an individual's social power to be close to her social influence while minimizing the so called "extropy" -- the entropy of the complimentary opinion.