Optimizing Leader Influence in Networks through Selection of Direct Followers

TL;DR

This paper addresses the problem of selecting direct followers for a leader in a network to maximize influence, introducing two heuristic methods with proven approximation guarantees and demonstrating high effectiveness in numerical tests.

Contribution

It introduces a unified combinatorial optimization framework for leader influence with new supermodularity and convexity results, and develops two heuristic algorithms with theoretical guarantees.

Findings

Greedy approach achieves approximation ratio better than (1-1/e)

Convex relaxation enables efficient numerical solutions

Numerical results show influence reach of 90% or higher

Abstract

The paper considers the problem of a leader that seeks to optimally influence the opinions of agents in a directed network through connecting with a limited number of the agents ("direct followers"), possibly in the presence of a fixed competing leader. The settings involving a single leader and two competing leaders are unified into a general combinatoric optimization problem, for which two heuristic approaches are developed. The first approach is based on a convex relaxation scheme, possibly in combination with the $\ell_1$-norm regularization technique, and the second is based on a greedy selection strategy. The main technical novelties of this work are in the establishment of supermodularity of the objective function and convexity of its continuous relaxation. The greedy approach is guaranteed to have a lower bound on the approximation ratio sharper than $(1-1/e)$, while the convex approach can benefit from efficient (customized) numerical solvers to have practically comparable solutions possibly with faster computation times. The two approaches can be combined to provide improved results. In numerical examples, the approximation ratio can be made to reach $90\%$ or higher depending on the number of direct followers.