Minimizing Polarization in Noisy Leader-Follower Opinion Dynamics

TL;DR

This paper addresses the problem of reducing opinion polarization in noisy social networks by adding edges, proposing algorithms with theoretical guarantees and demonstrating their effectiveness through experiments.

Contribution

It introduces a supermodular optimization framework for minimizing effective resistance in opinion dynamics, along with efficient approximation algorithms.

Findings

The proposed greedy algorithm achieves a $(1-1/e)$ approximation ratio.

The fast algorithm computes approximate effective resistance efficiently.

Experiments confirm the algorithms' effectiveness and efficiency.

Abstract

The operation of creating edges has been widely applied to optimize relevant quantities of opinion dynamics. In this paper, we consider a problem of polarization optimization for the leader-follower opinion dynamics in a noisy social network with $n$ nodes and $m$ edges, where a group $Q$ of $q$ nodes are leaders, and the remaining $n-q$ nodes are followers. We adopt the popular leader-follower DeGroot model, where the opinion of every leader is identical and remains unchanged, while the opinion of every follower is subject to white noise. The polarization is defined as the steady-state variance of the deviation of each node's opinion from leaders' opinion, which equals one half of the effective resistance $\mathcal{R}_Q$ between the node group $Q$ and all other nodes. Concretely, we propose and study the problem of minimizing $\mathcal{R}_Q$ by adding $k$ new edges with each incident to a node in $Q$. We show that the objective function is monotone and supermodular. We then propose a simple greedy algorithm with an approximation factor $1-1/e$ that approximately solves the problem in $O((n-q)^3)$ time. To speed up the computation, we also provide a fast algorithm to compute $(1-1/e-\eps)$-approximate effective resistance $\mathcal{R}_Q$, the running time of which is $\Otil (mk\eps^{-2})$ for any $\eps>0$, where the $\Otil (\cdot)$ notation suppresses the ${\rm poly} (\log n)$ factors. Extensive experiment results show that our second algorithm is both effective and efficient.