Maximizing Influence of Leaders in Social Networks

TL;DR

This paper studies how adding edges to social networks can maximize the influence of leaders with opposing opinions, proposing efficient algorithms with strong approximation guarantees and demonstrating scalability to large networks.

Contribution

It introduces a submodular optimization framework for edge addition to enhance leader influence, along with fast approximation algorithms with proven guarantees.

Findings

The objective function is monotone and submodular.

A greedy algorithm achieves a (1 - 1/e) approximation in polynomial time.

The second algorithm scales efficiently to networks with over a million nodes.

Abstract

The operation of adding edges has been frequently used to the study of opinion dynamics in social networks for various purposes. In this paper, we consider the edge addition problem for the DeGroot model of opinion dynamics in a social network with $n$ nodes and $m$ edges, in the presence of a small number $s \ll n$ of competing leaders with binary opposing opinions 0 or 1. Concretely, we pose and investigate the problem of maximizing the equilibrium overall opinion by creating $k$ new edges in a candidate edge set, where each edge is incident to a 1-valued leader and a follower node. We show that the objective function is monotone and submodular. We then propose a simple greedy algorithm with an approximation factor $(1-\frac{1}{e})$ that approximately solves the problem in $O(n^3)$ time. Moreover, we provide a fast algorithm with a $(1-\frac{1}{e}-\epsilon)$ approximation ratio and $\tilde{O}(mk\epsilon^{-2})$ time complexity for any $\epsilon>0$, where $\tilde{O}(\cdot)$ notation suppresses the ${\rm poly} (\log n)$ factors. Extensive experiments demonstrate that our second approximate algorithm is efficient and effective, which scales to large networks with more than a million nodes.